FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Fuzzy Topological Systems

Apostolos Syropoulos1113578 Valeria de Paiva1113579

1113578Greek Molecular Computing GroupXanthi Greece

asyropoulosgmailcom1113579School of Computer ScienceUniversity of Birmingham UKvaleriadepaivagmailcom

Segundo Congresso Brasileiro de Sistemas FuzzyNovember 6ndash9 2012

Natal Rio Grande do Norte Brazil

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Outline

1 VaguenessGeneral IdeasMany-valued Logics and Vagueness

2 Dialectica Spaces

3 Fuzzy Topological Spaces

4 Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline cases

What is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline case

A case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Outline

1 VaguenessGeneral IdeasMany-valued Logics and Vagueness

2 Dialectica Spaces

3 Fuzzy Topological Spaces

4 Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline cases

What is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline case

A case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline cases

What is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline case

A case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline cases

What is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline case

A case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline case

A case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague term

The adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vagueterm

ExamplesSerena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tall

Is Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge star

If you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated him

Eubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites Paradox

The old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

What is Vagueness

It is widely accepted that a term is vague to the extentthat it has borderline casesWhat is a borderline caseA case in which it seems possible either to apply or notto apply a vague termThe adjective ldquotallrdquo is a typical example of a vaguetermExamples

Serena is 175m tall is she tallIs Betelgeuse a huge starIf you cut one head off of a two headed man have youdecapitated himEubulides of Miletus The Sorites ParadoxThe old puzzle about bald people

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-many

Example A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale map

Charles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminate

Max Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Vagueness

Bertrand Russell Per contra a representation isvague when the relation of the representingsystem to the represented system is notone-one but one-manyExample A small-scale map is usually vaguer than alarge-scale mapCharles Sanders Peirce A proposition is vaguewhen there are possible states of thingsconcerning which it is intrinsically uncertainwhether had they been contemplated by thespeaker he would have regarded them asexcluded or allowed by the proposition Byintrinsically uncertain we mean not uncertain inconsequence of any ignorance of the interpreterbut because the speakerrsquos habits of languagewere indeterminateMax Black demonstrated this definition using the wordchair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notions

A term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in context

Example He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couch

Bertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquo

Example Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Vagueness Generality and Ambiguity

Vagueness ambiguity and generality are entirelydifferent notionsA term or phrase is ambiguous if it has at least twospecific meanings that make sense in contextExample He ate the cookies on the couchBertrand Russell ldquoA proposition involving a generalconceptmdasheg lsquoThis is a manrsquomdashwill be verified by anumber of facts such as lsquoThisrsquo being Brown or Jones orRobinsonrdquoExample Again consider the word chair

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)

Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquo

Jan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingencies

Emil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2

CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logics

In 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Many-valued Logics and Fuzzy Set Theory

Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione)discussed the problem of future contingencies (iewhat is the truth value of a proposition about a futureevent)Can someone tell us what is the truth value of theproposition ldquoAt Christmas there will be snow in AthensGreecerdquoJan Łukasiewicz defined a three-valued logic to dealwith future contingenciesEmil Leon Post introduced the formulation of 119899 truthvalues where 119899 ge 2CC Chang proposed MV-algebras that is an algebraicmodel of many-valued logicsIn 1965 Lotfi A Zadeh introduced his fuzzy settheory and its accompanying fuzzy logic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematics

Categories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystems

Dialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logic

Dialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri nets

Recent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Categorical Model of Linear Logic

Categories form a very high-level abstract mathematicaltheory that unifies all branches of mathematicsCategories have been used to model and study logicalsystemsDialectica spaces which were introduced by de Paivaform a category which is a model of linear logicDialectica categories have been used to model Petrinets the Lambek Calculus state in programming andto define fuzzy petri netsRecent development fuzzy topological systems that isthe fuzzy counterpart of Vickersrsquos topological systems

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if

(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset

∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure and

for any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Lineales

A quintuple (119871 le ∘ 1⊸) is a lineale if(119871 le) is a poset∘ ∶ 119871 times 119871 rarr 119871 is an order-preserving multiplication suchthat (119871 ∘ 1113568) is a symmetric monoidal structure andfor any 119886 119887 isin 119871 exists the largest 119909 isin 119871 such that 119886 ∘ 119909 le 119887then this element is denoted 119886 ⊸ 119887 and is called thepseudo-complement of 119886 with respect to 119887

The quintuple (I le and 1rArr) where I is the unit interval119886 and 119887 = 119886 119887 and 119886 rArr 119887 = ⋁119888 ∶ 119888 and 119886 le 119887(119886 or 119887 = 119886 119887) is a lineale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853)

Objects Triples 119860 = (119880 119883 120572) where 119880 and 119883 are setsand 120572 is a map 119880 times 119883 rarr I where I = [0 1]

Arrows An arrow from 119860 = (119880 119883 120572) to 119861 = (119881 119884 120573) is apair of 119826119838119853 maps (119891 119892) 119891 ∶ 119880 rarr 119881 119892 ∶ 119884 rarr 119883such that

120572(119906 119892(119910)) le 120573(119891(119906) 119910)

or in pictorial form

119880 times 119884id119880 times 119892- 119880 times 119883

ge

119881 times 119884

119891 times id119884

120573- I

120572

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

The Category 1202271202581202501202611113690(119826119838119853) cont

Arrow Composition Let (119891 119892) and (119891prime 119892prime) be the followingarrows

(119880 119883 120572)(119891119892)⟶ (119881119884 120573)

(119891prime119892prime)⟶ (119882119885 120574)

Then (119891 119892) ∘ (119891prime 119892prime) = (119891 ∘ 119891prime 119892prime ∘ 119892) such that

1205721114101119906 1114100119892prime ∘ 1198921114103(119911)1114104 le 12057411141011114100119891 ∘ 119891prime1114103(119906) 1199111114104

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)

Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 isthe relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Logical Connectives

Given objects 119860 = (119880 119883 120572) and 119861 = (119881 119884 120573)Tensor Product 119860otimes 119861 = (119880 times119881 119883119881 times119884119880 120572 times 120573) where 120572 times 120573 is

the relation that using the lineale structure ofI takes the minimum of the membershipdegrees

Linear Function-Space 119860 rarr 119861 = (119881119880 times 119884119883 119880 times 119883 120572 rarr 120573)where again the relation 120572 rarr 120573 is given by theimplication in the lineale

Product 119860 times 119861 = (119880 times 119881 119883 + 119884 120574) where120574 ∶ 119880 times 119881 times (119883 + 119884) rarr I is the fuzzy relation thatis defined as follows

1205741114100(119906 119907) 1199111114103 = 1114108120572(119906 119909) if 119911 = (119909 0)120573(119907 119910) if 119911 = (119910 1)

Coproduct 119860 oplus 119861 = (119880 + 119881 119883 times 119884 120575) where 120575 is definedsimilar to 120574

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Topological Systems

A triple (119880 119883 ⊧) where 119883 is a frame and 119880 is a set is atopological system if

when 119878 is a finite subset of 119883 then

119906 ⊧ 1114002119878 ⟺ 119906 ⊧ 119909 for all 119909 isin 119878

when 119878 is any subset of 119883 then

119906 ⊧ 1114004119878 ⟺ 119906 ⊧ 119909 for some 119909 isin 119878

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Defining Fuzzy Topological Systems

A triple (119880 119883 120572) where 119883 is a frame 119880 is a set and120572 ∶ 119880 times 119883 rarr I a binary fuzzy relation is a fuzzy topologicalsystem if

when 119878 is a finite subset of 119883 then

120572(1199061114002119878) le 120572(119906 119909) for all 119909 isin 119878

when 119878 is any subset of 119883 then

120572(1199061114004119878) le 120572(119906 119909) for some 119909 isin 119878

120572(119906 ⊤) = 1 and 120572(119906 perp) = 0 for all 119906 isin 119880

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852

Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results

The collection of objects of 1202271202581202501202611113716(119826119838119853) that are fuzzytopological systems and the arrows between them formthe category 119813119827119848119849119826119858119852119853119838119846119852Any topological system (119880 119883 ⊧) is a fuzzy topologicalsystem (119880 119883 120580) where

120580(119906 119909) = 11141081 when 119906 ⊧ 1199090 otherwise

The category of topological systems is a fullsubcategory of 1202271202581202501202611113716(119826119838119853)

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Results cont

Assume that 119886 isin 119860 where (119880 119883 120572) is a fuzzy topologicalspace Then the extent of an open 119909 is a functionwhose graph is given below

11141071114100119906 120572(119906 119909)1114103 ∶ 119906 isin 1198801114110

Now the collection of all fuzzy sets created by theextents of the members of 119860 correspond to a fuzzytopology on 119883

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Letrsquos be practical

Physical Interpratation 119880 set of programs that generate bitstream and elements of 119883 are assertions aboutbit streams

Example If 119906 generates the bit stream 010101010101hellipand ldquostarts 01010rdquo isin 119883 then119909 ⊧ starts 01010

Fuzzy Bit Streams Assume that 119909prime is a program thatproduces bit streams like the following

0 1 0 1 0 1 0 1 0

Explanation The bits above are distorted because of someinteraction with the environment

Result 119909prime satisfies the assertion ldquostarts 01010rdquo tosome degree

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquo

According to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909

When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909

When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909

This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

An alternative interpretation

Bart Kosko has argued that fuzziness ldquomeasures thedegree to which an event occurs not whether it occursRandomness describes the uncertainty of eventoccurrencerdquoAccording to this thesis expression 120572(119906 119909) would denotethe degree to which program 119906 will output a bitstreamthat will satisfy assertion 119909When 120572(119906 119909) = 0 119906 will not generate a bitstream thatsatisfies 119909When 120572(119906 119909) = 1 119906 will definitely generate a bistreamthat satisfies 119909This new physical interpretation is based on theassumption that the programs behave dynamically andnondeterministic

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presented

The notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vagueness

Dialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categories

Fuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systems

A ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale

FuzzyTopologicalSystems

Syropoulosde Paiva

VaguenessGeneral Ideas

Many-valued Logicsand Vagueness

DialecticaSpaces

FuzzyTopologicalSpaces

Finale

Finale

We have briefly presentedThe notion of vaguenessDialectica categoriesFuzzy topological systemsA ldquoreal liferdquo application of fuzzy topological systems

Thank you so much for your attention

• Vagueness
• • General Ideas
  • Many-valued Logics and Vagueness
  • • Dialectica Spaces
    • Fuzzy Topological Spaces
    • Finale