Problems of Proof Theory in Modal Logic - Portal · Questions for a proof theory of modal logic Proof analysis in modal logic First-order modal logic Completeness for Kripke semantics

Questions for a proof theory of modal logicProof analysis in modal logic

First-order modal logicCompleteness for Kripke semantics

Other non-classical logicsDoes the deduction theorem fail for modal logic?

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Problems of Proof Theory in Modal Logic

Sara Negri

University of Helsinki

Workshop on Recent Trends in Proof TheoryUniversity of Bern, July 9–11, 2008

Sara Negri Problems of Proof Theory in Modal Logic




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“The omission of proof theory and automated reasoning techniquescalls for a little more explanation. ... as is often the case in modallogic, the proof systems discussed are basically Hilbert-styleaxiomatic systems. There is no discussion of natural deduction,sequent calculi, labelled deductive systems, resolution, or displaycalculi. ... Why is this? Essentially because modal proof theoryand automated reasoning are still relatively youthful enterprises;they are exciting and active fields, but yet there is little consensusabout methods and few general results.”Blackburn, de Rijke, and Venema (2001, p. xvi).





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I Does the deduction theorem hold for modal logic?Deduction theorem: If Γ,A ` B, then Γ ` A ⊃ B.By the necessitation rule A ` 2A, but A ⊃ 2A is not atheorem.

I Can there be a sequent calculus for modal logic?

I How general a proof theory for modal logic can be developed?





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Aims and motivationsAxiomatic modal logicKripke semanticsThe method of proof analysisThe systems G3K and G3K∗

I Aim: Modular development of sequent calculi for a wide classof modal logics

I Requirements:I structural rules admissibleI simple syntaxI support, whenever possible, proof searchI calculi are uniform, thus allow easy proofs of meta-theoretic

results (conservativity, undefinability, etc.)I direct and uniform proof of completeness





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I Aim: Modular development of sequent calculi for a wide classof modal logics

I Requirements:I structural rules admissibleI simple syntaxI support, whenever possible, proof searchI calculi are uniform, thus allow easy proofs of meta-theoretic

results (conservativity, undefinability, etc.)I direct and uniform proof of completeness





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Basic modal logic K: Add to classical propositional logic:

1. 2(A ⊃ B) ⊃ (2A ⊃ 2B),2. From A to infer 2A.

Extensions of basic modal logic:Axiom

T 2A ⊃ A

4 2A ⊃ 22A

E 3A ⊃ 23A

B A ⊃ 23A

3 2(2A ⊃ B) ∨2(2B ⊃ A)

D 2A ⊃ 3A

2 32A ⊃ 23A

W 2(2A ⊃ A) ⊃ 2A





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I Kripke semantics (relational semantics) from the late fifties(S5 - Kripke 1959, other modal logics - Kripke 1963,intuitionistic logic - Kripke 1965).

I Had significant anticipations in the work of Arnould Bayart,Rudolf Carnap, Jaakko Hintikka, Stig Kanger, RichardMontague, Arthur Prior, and others (see Goldblatt 2005).

I Kripke frame: a set W of possible worlds, and a binaryrelation R between elements of W , accessibility relation

I Kripke model: Kripke frame with a valuationval : W × AtFrm −→ {0, 1}val(w ,P) = 1 P true at world ww P w forces P.





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Valuations are extended in a unique way to arbitrary formulas bymeans of inductive clausesw A&B whenever w A and w B,w A ∨ B whenever w A or w B,w A ⊃ B whenever from w A follows w B,w ⊥ for no w ,w 2A if and only if for all o, o A follows from wRo.





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Key insight of Kripke semantics: Axioms of different systems ofmodal logic correspond to special properties of the accessibilityrelation

Axiom Frame propertyT 2A ⊃ A ∀w wRw reflexivity4 2A ⊃ 22A ∀wor(wRo & oRr ⊃ wRr) transitivityE 3A ⊃ 23A ∀wor(wRo & wRr ⊃ oRr)

euclideannessB A ⊃ 23A ∀wo(wRo ⊃ oRw) symmetry3 2(2A ⊃ B) ∨2(2B ⊃ A) ∀wor(wRo & wRr ⊃ oRr ∨ rRo)

connectedness

D 2A ⊃ 3A ∀w∃o wRo seriality2 32A ⊃ 23A ∀wor(wRo & wRr ⊃ ∃l(oRl & rRl))

directedness

W 2(2A ⊃ A) ⊃ 2A no infinite R-chains + transitivity





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I Method: Based on a systematic development of translationof axioms into rules to be added to sequent calculus

I Applications:I extensions of logic, i.e. mathematical theories such as lattice

theory, real closed fields, affine and projective geometry,geometric and cogeometric theories (N, von Plato)intuitionistic theories of apartness and equality (Boretti, N)

I non-classical logics through labelled systems: modal logic,provability logic, intermediate logics, relevance, substructurallogics

I multimodal systems:

I formal epistemology: logic of common knowledge, logic ofdistributed knowledge (Hakli, N)

I temporal logic (Boretti, N)





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Background on labelled modal systems

Method for formulating systems of sequent calculus for basicmodal logic and its extensions and for systems of non-classicallogics in Negri (2005a,b).General ideas for basic modal logic K and other systems of normalmodal logics.Syntax embodies the possible world semantics:

I Rules for 2, 3 through meaning explanation and inversionprinciples.

I Frame properties in the form of “non-logical rules” added tothe basic sequent calculus. Extensions are modular.





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Sequent calculus for the basic modal logic K

I Start with the calculus G3c for classical propositional logic

I Enrich the language: Sequents Γ ⇒ ∆ where Γ and ∆ consistof expressions wRo and w : A (corresponding to x A ofKripke models), with w , o, r , l , ... ranging in a set W and withA any formula in the language of propositional logic extendedwith the modal operators of necessity and possibility, 2 and3.

I Rules for basic modal logic obtained from the inductivedefinition of validity in a Kripke frame.





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Fromw : 2A iff for all o, wRo implies o : A

obtain the rules

I If o : A can be derived for an arbitrary o accessible from w ,then w : 2A can be derived

wRo, Γ ⇒ ∆, o : A

Γ ⇒ ∆,w : 2AR2

arbitrariness of o becomes the variable condition o not (free)in Γ,∆

I If w : 2A and o is accessible from w , then o : A

o : A,w : 2A,wRo, Γ ⇒ ∆

w : 2A,wRo, Γ ⇒ ∆L2





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Fromw : 2A iff for all o, wRo implies o : A

obtain the rules

I If o : A can be derived for an arbitrary o accessible from w ,then w : 2A can be derived

wRo, Γ ⇒ ∆, o : A

Γ ⇒ ∆,w : 2AR2

arbitrariness of o becomes the variable condition o not (free)in Γ,∆

I If w : 2A and o is accessible from w , then o : A

o : A,w : 2A,wRo, Γ ⇒ ∆

w : 2A,wRo, Γ ⇒ ∆L2





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The rules for 3 are obtained similarly from the semanticalexplanation

w : 3A iff for some o, wRo and o : A

I If w : 3A, there exists o is accessible from w such that o : A

wRo, o : A, Γ ⇒ ∆

w : 3A, Γ ⇒ ∆L3

variable condition: o not (free) in Γ,∆

I If o is accessible from w and o : A, then w : 3Agives the rule

wRo, Γ ⇒ ∆,w : 3A, o : A

wRo, Γ ⇒ ∆,w : 3AR3





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The rules for 3 are obtained similarly from the semanticalexplanation

w : 3A iff for some o, wRo and o : A

I If w : 3A, there exists o is accessible from w such that o : A

wRo, o : A, Γ ⇒ ∆

w : 3A, Γ ⇒ ∆L3

variable condition: o not (free) in Γ,∆

I If o is accessible from w and o : A, then w : 3Agives the rule

wRo, Γ ⇒ ∆,w : 3A, o : A

wRo, Γ ⇒ ∆,w : 3AR3





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Initial sequents:

w : P, Γ ⇒ ∆, w : P wRo, Γ ⇒ ∆, wRo

Propositional rules:

w : A, w : B, Γ ⇒ ∆

w : A&B, Γ ⇒ ∆L&

Γ ⇒ ∆, w : A Γ ⇒ ∆, w : B

Γ ⇒ ∆, w : A&BR&

w : A, Γ ⇒ ∆ w : B, Γ ⇒ ∆

w : A ∨ B, Γ ⇒ ∆L∨

Γ ⇒ ∆, w : A, w : B

Γ ⇒ ∆, w : A ∨ BR∨

Γ ⇒ ∆, w : A w : B, Γ ⇒ ∆

w : A ⊃ B, Γ ⇒ ∆L⊃

w : A, Γ ⇒ ∆, w : B

Γ ⇒ ∆, w : A ⊃ BR⊃

w :⊥, Γ ⇒ ∆L⊥

Modal rules:

o : A, w : 2A, wRo, Γ ⇒ ∆

w : 2A, wRo, Γ ⇒ ∆L2

wRo, Γ ⇒ ∆, o : A

Γ ⇒ ∆, w : 2AR2

wRo, o : A, Γ ⇒ ∆

w : 3A, Γ ⇒ ∆L3

wRo, Γ ⇒ ∆, w : 3A, o : A

wRo, Γ ⇒ ∆, w : 3AR3





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Modal systems and frame properties

I Modal logic K characterized by arbitrary frames.

I Restrictions of the class of frames amount to adding certainframe properties to the calculus.

I In Kripke frames for S4 the accessibility relation is reflexive∀wwRwand transitive∀w∀o∀r((xwRo & oRr) ⊃ wRr)

I For S4.2 also directed ∀wor(wRo & wRr ⊃ ∃l(oRl & rRl))





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Modal systems and frame properties

I Modal logic K characterized by arbitrary frames.

I Restrictions of the class of frames amount to adding certainframe properties to the calculus.

I In Kripke frames for S4 the accessibility relation is reflexive∀wwRwand transitive∀w∀o∀r((xwRo & oRr) ⊃ wRr)

I For S4.2 also directed ∀wor(wRo & wRr ⊃ ∃l(oRl & rRl))





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Axioms as rules (cont.)

I We use the method developed in NvP (1998) and in N (2003)for extending sequent calculi with rules for axiomatic theorieswhile preserving all the structural properties of the logicalcalculus.

I Universal axioms are transformed into conjunctive normalform, that is, conjunctions of formulas of the formP1& . . .&Pm ⊃ Q1 ∨ · · · ∨ Qn

I Each conjunct is then converted into the regular rule,

Q1, Γ ⇒ ∆ . . . Qn, Γ ⇒ ∆

P1, . . . ,Pm, Γ ⇒ ∆Reg

Omitted details: repetion of the Pi ’s in the premisses andclosure condition.





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Q1, Γ ⇒ ∆ . . . Qn, Γ ⇒ ∆

P1, . . . ,Pm, Γ ⇒ ∆Reg






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Q1, Γ ⇒ ∆ . . . Qn, Γ ⇒ ∆

P1, . . . ,Pm, Γ ⇒ ∆Reg






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I Method extended (N2003) to geometric theories, that is,theories axiomatized by formulas

∀z(A ⊃ B)

where A and B do not contain ⊃ or ∀.

I These can be reduced to conjunctions of

∀z(P1& . . .&Pm ⊃ ∃x1M1 ∨ · · · ∨ ∃xnMn)

where Mj is conjunction of atoms Qj

I and turned into the geometric rule, with the yi ’s not in theconclusion

Q1(y1/x1),P, Γ ⇒ ∆ . . . Qn(yn/xn),P, Γ ⇒ ∆

P, Γ ⇒ ∆GR

I example: ∀x∃yxRy (seriality)





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∀z(A ⊃ B)

where A and B do not contain ⊃ or ∀.I These can be reduced to conjunctions of

∀z(P1& . . .&Pm ⊃ ∃x1M1 ∨ · · · ∨ ∃xnMn)



Q1(y1/x1),P, Γ ⇒ ∆ . . . Qn(yn/xn),P, Γ ⇒ ∆

P, Γ ⇒ ∆GR

I example: ∀x∃yxRy (seriality)





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∀z(A ⊃ B)

where A and B do not contain ⊃ or ∀.I These can be reduced to conjunctions of

∀z(P1& . . .&Pm ⊃ ∃x1M1 ∨ · · · ∨ ∃xnMn)



Q1(y1/x1),P, Γ ⇒ ∆ . . . Qn(yn/xn),P, Γ ⇒ ∆

P, Γ ⇒ ∆GR

I example: ∀x∃yxRy (seriality)Sara Negri Problems of Proof Theory in Modal Logic




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Structural properties of the basic system

I Theorem The structural rules are admissible in extensions ofG3c with regular or geometric rules that satisfy the closurecondition. Weakening and contraction are height-preservingadmissible. All the rules are invertible.

I The same structural properties hold in G3K, in additionsubstitution of labels is height-preserving admissible.





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Structural properties of the basic system

I Theorem The structural rules are admissible in extensions ofG3c with regular or geometric rules that satisfy the closurecondition. Weakening and contraction are height-preservingadmissible. All the rules are invertible.

I The same structural properties hold in G3K, in additionsubstitution of labels is height-preserving admissible.





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Structural properties of extensions

I Let G3K* be any extension of G3K with rules for theaccessibility relation.

I All the structural rules–weakening, contraction, and cut–areadmissible in the system G3K*.

I The characteristic axioms are derivable.

I The necessitation rule is admissible.

I The rules are sound wrt Kripke semantics.

I Answers to questions of undefinability though conservativitytheorems.

I Answers to decidability questions through algorithms ofterminating proof search.





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Quantificational modelRules for the quantifiersBarcan and the likeStructural properties of G3Kq*

Quantificational model (Kripke 1963): To every world w isassociated a domain of interpretation of individual variables D(w).D ≡

⋃w∈K D(w)

Valuation of n-ary predicates P(x1, . . . , xn) is a function [[P]] fromDn to the set of classical truth values {0, 1}w forces P(x1, . . . , xn) under the assignmentσ ≡ 〈a1/x1, . . . an/xn〉, w σ P(x1, . . . , xn), if [[P]](a1, . . . , an) = 1.Different possible assumptions about the domains D(w) give riseto different notions of quantificational models.We proceed with a formal analysis of the notion to be importedinto our syntactic treatment.Interaction of modalities with the proper quantifiers captured byconditions that connect possible worlds and ranges of variables.





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Given a valuation of atomic predicates under an assignment, it isextended into a valuation of arbitrary formulas in a quantificationalmodel by the standard inductive clauses for propositionalconnectives. For the quantifiers

w σ ∀xA(x , y1, . . . , yn) wheneverfor all σ′ such that σ′(yi ) = σ(yi ), w σ′ ∀xA(x , y1, . . . , yn).

By leaving arbitrary the assignment of free variables

w ∀xA(x) whenever for all a in D(w), w A(a/x).

w ∃xA(x) whenever for some a in D(w), w A(a/x).

We add to our calculus expressions of the form a ∈ D(w) that canappear in sequents together with labelled formulas w : A andrelational atoms wRo





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Quantifier rules of G3Kq

a ∈ D(w), Γ ⇒ ∆,w : A(a/x)

Γ ⇒ ∆,w : ∀xA R∀

w : A(a/x),w : ∀xA, a ∈ D(w), Γ ⇒ ∆

w : ∀xA, a ∈ D(w), Γ ⇒ ∆L∀

a ∈ D(w), Γ ⇒ ∆,w : ∃xA,w : A(a/x)

a ∈ D(w), Γ ⇒ ∆,w : ∃xA R∃

a ∈ D(w),w : A(a/x), Γ ⇒ ∆

w : ∃xA, Γ ⇒ ∆L∃

Rules R∀, L∃ have the condition a /∈ Γ,∆.Sara Negri Problems of Proof Theory in Modal Logic




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In axiomatic approaches to modal logic (e.g. Hughes & Cresswell)first-order modal logic obtained as an extension of first-orderclassical logic;This leads to a mismatch between axiomatization and semantics,for example the principle of universal instantiation, in the form∀xA(x) ⊃ A(a) in general is not valid.This is seen through a failed proof search by our rules:

....w : ∀xA(x) ⇒ w : A(y)

⇒ w : (∀xA(x) ⊃ A(a))R⊃

After this single step, no rule is applicable; The only way tocontinue would be a step of L∀ but this would require theadditional assumption a ∈ D(w). By the completeness of thecalculus, the proof search is exhaustive.





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Similarly to G3K, we may add to G3Kq properties of theaccessibility relation and obtain, for example, a system for S5 withquantifiers by adding rules Ref, Trans, and Sym.Also properties of the domain function can be required. Forinstance, it can be postulated that for every world, thecorresponding domain of interpretation be non-empty:

∀w∃a(a ∈ D(w))

Another condition is that domains are increasing:

∀wo∀a(wRo & a ∈ D(w) ⊃ a ∈ D(o))

They can also be decreasing:

∀wo∀a(wRo & a ∈ D(o) ⊃ a ∈ D(w))





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All the above properties follow the geometric rule scheme and theirrule form is a follows, with the variable condition a /∈ Γ,∆ in thefirst:

a ∈ D(w), Γ ⇒ ∆

Γ ⇒ ∆Nonempty

a ∈ D(o),wRo, a ∈ D(w), Γ ⇒ ∆

wRo, a ∈ D(w), Γ ⇒ ∆Incr

a ∈ D(w),wRo, a ∈ D(o), Γ ⇒ ∆

wRo, a ∈ D(o), Γ ⇒ ∆Decr





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Property of nonemptiness usually part of the ontology of theintended semantics for quantified systems of logic and implicit inthe rule of elimination of the universal quantifier. We gain a moreflexible approach by not having it inbuilt in the rules. Formallysimilar to the property of seriality added to G3K to obtain deonticlogic. This latter characterized by the axiom 2A ⊃ 3A.Nonemptiness corresponds to the axiom ∀xA ⊃ ∃xA

w : A(a/x), a ∈ D(w),w : ∀xA ⇒ w : ∃xA,w : A(a/x)

w : A(a/x), a ∈ D(w),w : ∀xA ⇒ w : ∃xA R∃

a ∈ D(w),w : ∀xA ⇒ w : ∃xA L∀

w : ∀xA ⇒ w : ∃xANonempty

⇒ w : ∀xA ⊃ ∃xA R⊃





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Properties of permutability of the necessity modality and theuniversal quantifier have been the object of a long philosophicaldiscussion.Barcan formula ∀x2A ⊃ 2∀xAConverse Barcan formula 2∀xA ⊃ ∀2xA.The Barcan formula is derivable in G3Kq+Decr

o : A(a/x),w : 2A(a/x), a ∈ D(w), a ∈ D(o),wRo,w : ∀x2A ⇒ o : A(a/x)

w : 2A(a/x), a ∈ D(w), a ∈ D(o),wRo,w : ∀x2A ⇒ o : A(a/x)L2

a ∈ D(w), a ∈ D(o),wRo,w : ∀x2A ⇒ o : A(a/x)L∀

a ∈ D(o),wRo,w : ∀x2A ⇒ o : A(a/x)Decr

wRo,w : ∀x2A ⇒ o : ∀xA R∀

w : ∀x2A ⇒ w : 2∀xA R2

⇒ w : ∀x2A ⊃ 2∀xA R⊃





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The converse Barcan formula is derivable in G3Kq+Incr

o : A(a/x), o : ∀xA, a ∈ D(o),wRo, a ∈ D(w),w : 2∀xA ⇒ o : A(a/x)

o : ∀xA, a ∈ D(o),wRo, a ∈ D(w),w : 2∀xA ⇒ o : A(a/x)L∀

a ∈ D(o),wRo, a ∈ D(w),w : 2∀xA ⇒ o : A(a/x)L2

wRo, a ∈ D(w),w : 2∀xA ⇒ o : A(a/x)Incr

a ∈ D(w),w : 2∀xA ⇒ w : 2A(a/x)R2

w : 2∀xA ⇒ ∀x2AR∀

⇒ w : 2∀xA ⊃ ∀x2AR⊃





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All the structural properties proved for G3Kq* easily extend toG3Kq*.In addition to substitution on labels we have substitution ondomain elements:

a ∈ D(w)(b/a) ≡ b ∈ D(w)a ∈ D(w)(o/w) ≡ a ∈ D(o)

In addition to the weakening and contraction rules of G3K, wehave to consider also weakening and contraction rules that operateon domain atoms, such as

Γ ⇒ ∆x ∈ D(w), Γ ⇒ ∆

LWDa ∈ D(w), a ∈ D(w), Γ ⇒ ∆

a ∈ D(w), Γ ⇒ ∆L-CtrD





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I If Γ ⇒ ∆ is derivable in G3Kq*, then alsoΓ(o/w) ⇒ ∆(o/w) and Γ(b/a) ⇒ ∆(b/a) are derivable, withthe same derivation height (substitution of world labels and ofdomain elements is heigh-preserving admissible).

I All the structural rules (weakening and contraction, both onlabelled formulas and on relational and domain atoms) areheight-preserving admissible in G3Kq*.

I The rule of necessitation is admissible in G3Kq*.

I The rule of cut is admissible in G3K*.





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History: The original proofHistory: Criticisms and their effectUniform completeness

I Kripke (1959) Completeness for 1st-order S5 with equality. Aformula is derivable in S5 if and only if it is valid using anadaptation to modal logic of Beth’s technique of semanticaltableaux. If A1& . . .&An ⊃ B is not valid, the tableauconstruction cannot be closed; If the tableau construction isnot closed, then a countermodel is found.

I Kripke (1963) Extension of results of Kripke (1959) to T, B,S4, and S5. Properties of the accessibility relation not part ofthe tableau syntax.Validity: If the construction for A is closed then A is valid.Completeness: By a systematic search of a countermodel.Uses Konig’s lemma; Applications to decidability.Modal disjunction property (McKinsey-Tarski 1948, Lemmon1960) here with “glueing of Kripke models”.





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I Kripke (1959) Completeness for 1st-order S5 with equality. Aformula is derivable in S5 if and only if it is valid using anadaptation to modal logic of Beth’s technique of semanticaltableaux. If A1& . . .&An ⊃ B is not valid, the tableauconstruction cannot be closed; If the tableau construction isnot closed, then a countermodel is found.

I Kripke (1963) Extension of results of Kripke (1959) to T, B,S4, and S5. Properties of the accessibility relation not part ofthe tableau syntax.Validity: If the construction for A is closed then A is valid.Completeness: By a systematic search of a countermodel.Uses Konig’s lemma; Applications to decidability.Modal disjunction property (McKinsey-Tarski 1948, Lemmon1960) here with “glueing of Kripke models”.





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I Arnould Bayart (1966), review of Kripke (1959): Criticizeslack of determinism in the tableau construction for thequantifier rules.

I David Kaplan (JSL, 1966), review to Kripke (1963):Criticizes lack of rigor and use of informal arguments in thetableau construction. Suggests a completeness proof a’ laHenkin, as foreseen already by Kanger, as more rigorous.

I Henkin-style completenessHenkin (1949) completeness for 1st-order logicBayart (1958,1959) Sequent calculus, possibleworld semantics, and Henkin sytle completenessfor 1st- and 2nd-order S5Lemmon, Scott (Kaplan 1966)Makinson (1966)Cresswell (1967)

Henkin-style completeness proof for modal logic:Soundness proved by induction of the derivation in L. If L hasadditional axioms then it is proved that they are valid in the classof frames considered.Completeness proved by the canonical model construction. From La special model is built in which validity and derivability coincide.The canonical model is a Kripke model in which the nodes aremaximal consistent sets of formulas, the accessibility relation issuch that two nodes Γ,∆ are related if all the necessary truth inthe former are in the latter and a formula is forced at a node if itbelongs to that node.





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I The use of a labelled system allows a direct proof ofcompleteness via Schutte’s method of reduction trees.

I The proof is not constructive (Konig’s lemma), however

I The countermodels of unprovable sequents are obtaineddirectly from a failed proof search

I No need for the somewhat artificial constructions of Henkinsets of formulas and of “bulldozing” methods for imposingirreflexivity (in systems with irreflexive relations).

I The proof can be adapted to all the systems ofmodal/temporal/epistemic logic considered.





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Uniform completeness: soundness

Consider a derivation in G3K∗:

K frame with accessibility relation R that satisfies the properties ∗.W the set of world labels used in derivations in G3K∗.

Interpretation of W in K ≡ [[·]] : W → K such that,

If wRo in the derivation, then [[w ]]R[[o]] in K .

Valuation of atomic formulas V : AtFrm → P(K )

w ∈ V(P) iff [[w ]] P.





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Uniform completeness: soundness (cont.)

Valuations extended to arbitrary formulas:

For all w , it is not the case that [[w ]] ⊥ (abbr.[[w ]] 1 ⊥);

[[w ]] A&B if [[w ]] A and [[w ]] B;

[[w ]] A ∨ B if [[w ]] A or [[w ]] B;

[[w ]] A ⊃ B if [[w ]] A implies [[w ]] B;

[[w ]] 2A if for all o, [[w ]]R[[o]] implies [[o]] A;

[[w ]] 3A if there exists o such that [[w ]]R[[o]] and[[o]] A.





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Uniform completeness: soundness (cont.)

Γ ⇒ ∆ valid for a given interpretation of labels and valuation ofpropositional variables in a frame, if for all labelled formulas w : Aand relational atoms oRr in Γ, if [[w ]] A and [[o]]R[[r ]] in K , thenfor some l : B in ∆, [[l ]] B. A sequent is valid if it it valid forevery interpretation and every valuation of propositional variablesin the frame.

Validity: If sequent Γ ⇒ ∆ is derivable in G3K∗, then it is valid.





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Uniform completeness: completeness

Completeness: Let Γ ⇒ ∆ be a sequent in the language ofG3K∗. Then either the sequent is derivable in G3K∗ or it has aKripke countermodel.Proof: We define for an arbitrary sequent Γ ⇒ ∆ in the languageof G3K∗ a reduction tree by applying root first the rules of G3K∗

in all possible ways. If the construction terminates we obtain aproof, else we obtain an infinite tree. By Konig’s lemma an infinitetree has an infinite branch, which is used to define a countermodelto the endsequent.





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Uniform completeness: completeness (cont.)

Construction of the countermodel:Let Γ0 ⇒ ∆0 ≡ Γ ⇒ ∆, Γ1 ⇒ ∆1 . . . , Γi ⇒ ∆i , . . . be the infinitebranch branch. Consider the sets of labelled formulas andrelational atoms

Γ ≡⋃i>0

Γi

∆ ≡⋃i>0

∆i

We define a Kripke model that forces all the formulas in Γ and noformula in ∆, and is therefore a countermodel to the sequentΓ ⇒ ∆.





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Uniform completeness: completeness (cont.)

Frame K ≡ labels appearing in the relational atoms in Γ

Relation R ≡ all the wRo’s in Γ.

Construction of the reduction tree imposes the frame properties ofthe countermodel: For instance, in the system G3S4 theconstructed frame is reflexive and transitive.

Valuation: For all the atomic formulas w : P in Γ, set w P, andfor all atomic formulas o : Q in ∆, set o 1 Q.

Finally show inductively on the weight of formulas that A is forcedin the model at node w if w : A is in Γ and A is not forced at nodew if w : A is in ∆. Therefore we have a countermodel to theendsequent Γ ⇒ ∆.





References

I Provability logic (N 2005): Noetherian frame conditions, notfirst-order, part of the modal rules.

I Intermediate propositional logics (Dyckhoff & N 2005):Uniform calculi and embeddings between the interpolableintermediate propositional logics , and the correspondingmodal logics between S4 and S5.

I Substructural logics (N 2008): Uniform treatment throughthe Routley-Meyer relational semantics, a generalization ofthe standard relational semantics for intuitionistic and modallogic: Instead of a binary accessibility relation, we have aternary relation R.

I Epistemic logic: Logic of distributed knowledge (Hakli & N2008), logic of common knowledge (wip).

I Tense logic: Linear time (Boretti & N 2008), branching time(wip)





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Does the deduction theorem fail for modal logic?

YES NO POSSIBLYBarcan-Marcus(1946,1953)

Zeman (1967) Feys (1965)Satre (1972) Zeman (1973)

Fagin et al. (1995) Chellas (1980) Sundholm (1983)Basin et al. (1998) Hughes & Cresswell (1996) Avron (1991)

Chagrov &Zakharyaschev (1997)

Ganguli & Nerode (2004) Fitting (2007)





References

I Deduction theorem: If Γ,A ` B, then Γ ` A ⊃ B.

I Standard argument for failure thesis: By the necessitation ruleA ` 2A, but A ⊃ 2A is not a theorem.

I Rule of necessitation originally formulated for Hilbert systems(all steps in a derivation are axioms or tautologies) need to berestricted when used in hypothetical derivations: If ` A, then` 2A. The problem vanishes because 2A does not followfrom the assumption A, i.e. A ` 2A is not derivable, in factthe unrestricted rule of necessitation is not even sound.





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I Hilbert system for the basic modal logic K:

A0. A ⊃ AA1. A ⊃ (B ⊃ A)A2. (A ⊃ (A ⊃ B)) ⊃ (A ⊃ B)A3. (A ⊃ (B ⊃ C )) ⊃ (B ⊃ (A ⊃ C ))A4. (B ⊃ C ) ⊃ ((A ⊃ B) ⊃ (A ⊃ C ))A5. ¬¬A ⊃ AB1. 2(A ⊃ B) ⊃ (2A ⊃ 2B)R1. From A and A ⊃ B infer B (modus ponens)R2. From A infer 2A (necessitation)

I Extension with assumptions HK:A ∈ ΓΓ ` A

A ∈ AxiomΓ ` A

Γ ` A ∆ ` A ⊃ BΓ,∆ ` B

` AΓ ` 2A





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I Hilbert system for the basic modal logic K:

A0. A ⊃ AA1. A ⊃ (B ⊃ A)A2. (A ⊃ (A ⊃ B)) ⊃ (A ⊃ B)A3. (A ⊃ (B ⊃ C )) ⊃ (B ⊃ (A ⊃ C ))A4. (B ⊃ C ) ⊃ ((A ⊃ B) ⊃ (A ⊃ C ))A5. ¬¬A ⊃ AB1. 2(A ⊃ B) ⊃ (2A ⊃ 2B)R1. From A and A ⊃ B infer B (modus ponens)R2. From A infer 2A (necessitation)

I Extension with assumptions HK:A ∈ ΓΓ ` A

A ∈ AxiomΓ ` A

Γ ` A ∆ ` A ⊃ BΓ,∆ ` B

` AΓ ` 2A





References

For the system HK we have:

I Monotonicity of derivability:If Γ ` A, then Γ′ ` A for any Γ′ that extends Γ.

I Deduction theorem: If A, Γ ` B, then Γ ` A ⊃ B.

I Closure under composition:If Γ′ ` A and A, Γ′′ ` B then Γ′, Γ′′ ` B.

I Principle of detachment:If Γ ` A ⊃ B, then A, Γ ` B.

I Full and faithful translation to a contraction- and cut-freesequent calculus for K shows that A ` 2A in general is notderivable, and that if it is derivable, also ` A ⊃ 2A is.





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References

I R. Dyckhoff and S. Negri (2005) Proof analysis in intermediatepropositional logics, ms.

I Negri S. (2003), Contraction-free sequent calculi for geometrictheories, with an application to Barr’s theorem, Archive forMathematical Logic, vol. 42, pp. 389-401.

I Negri S. (2005), Proof analysis in modal logic, Journal ofPhilosophical Logic, vol 34, pp. 507–544, 2005

I Negri, S. and J. von Plato (2001) Structural Proof Theory,Cambridge University Press.





References

I Boretti, B. and S. Negri (2007), On the Finitization ofPriorean Linear Time, ms.

I Hakli, R. and S. Negri (2008), Proof theory for distributedknowledge, Springer Lecture Notes in Artificial Intelligence, inpress.

I Hakli, R. and S. Negri (2008a), Does the deduction theoremfail for modal logic?, ms.

I Negri S. (2008), Proof analysis in non-classical logics, invitedpaper at Logic Colloquium 2005. C. Dimitracopoulos et al.(eds) ASL Lecture Notes in Logic, vol. 28, pp. 107-128.

I Negri, S. and J. von Plato (2008) Proof Analysis, bookmanuscript.


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Problems of Proof Theory in Modal Logic - Portal · Questions for a proof theory of modal logic Proof analysis in modal logic First-order modal logic Completeness for Kripke semantics