Many Valued Logic (MVL)

By: Shay Erov - 01/11/2007

Many Valued Logic

• Today’s lecture:– What is MVL?– Motivation– MVL structure– Some 3-valued logics– Consequence relations– n-valued logics– Dunn/Belnap 4-valued logic– Extension to the first-order level

Many Valued Logic

• Today’s lecture (continued) :– Alternative– Fuzzy Logic (∞-valued logics)– Applications– History review– Summary

What is MVL?

• A truth-functional logic containing more than 2 truth values

• We will mainly focus on 3-valued logics, and Fuzzy logics (∞-valued )

Motivation

• Motivation for Many Valued Logic came mainly from paradoxes raised by philosophers

• MVL usually comes to give an answer to questions where it is hard / impossible to determine the truth values

• In the next couple of slides we will cover some of those motivations

Motivation – Inconsistent law

• Let assume that in a certain country, the following rules apply:– No aborigine shall have the right to vote– All property holders shall have the right to

vote

• Now, let’s say John is an aborigine that did manage to buy a property. Is John allowed to vote?

Motivation – self reference

• Consider the following two paradoxes:– The Liar’s paradox : This sentence is false– Russell's paradox: Define a group r in the

following way: {x| x x}. is r r?

• Any truth value assigned to one of the above, will result in it having both true and false values.

• Extended paradox : this sentence is either false or neither true nor false

Motivation – denotation failure

• There may be a sentence where neither A nor A can be verified

• “The present king of Canada was born in Italy”

• “Sherlock Holmes had 3 aunts”

• Sentences containing noun references that do not refer to anything

• Frege suggested that all such sentences are neither true nor false. Is that right?

Motivation – Future Contingents

• The sea battle paradox– There will be a sea battle tomorrow

• If there’s a truth value, then what will happen is of necessity, and it’s not possible since it’s still contingent matter.

Motivation – Future Contingents

• Łukasiewicz : If statements about the future events are already true or false, then the future is as much determined as the past and differs from the past only in so far as it has not yet come to pass

MVL – Definitions

• Language (atomic propositions, Connectives)

• Assignment / Interpretation

• Valid formula

• Designated values

MVL – Structure

• • - Set of truth degrees• - Set of designated truth values• - for every n-place connective c, an

n-place function on V to denote it

• Truth degrees meaning

• How do we define the classical logic this way?

{ , ,{ | }cV D f c CVDcf

3-valued logics

• Consist of 3 truth values, 2 main types for the third value, i,:– Neither true nor false– Both true and false

• and now for some examples…

3-valued logics –

• Invented by Łukasiewicz • i stands for ‘possible’

• {1} as the only designated value

• The tables for are intuitive

• Why is also 1 ?

10

ii

01

f 1i0

11i0

iii0

0000

f 1i0

1111

i1ii

01i0

f 1i0

11i0

i11i

0111

f

3L

{ , , }

i i

3-valued logics –

• Invented by Kleene• i stands for ‘possible’

• {1} as the only designated value

• Same as besides

• What is the problem here?

10

ii

01

f 1i0

11i0

iii0

0000

f 1i0

1111

i1ii

01i0

f 1i0

11i0

i1ii

0111

f

3K

i i3L

3-valued logics –

• i stands for ‘both true and false’

• {1, i} as the designated values

• Tables the same as

• Follows the rule of excluded middle

• Does not follow Modus Ponens

LP

3K

p p

,p p q q

3-valued logics –

• Based on LP, change in table of

• Follows Modus Ponens

3RM

f

1i0

1100

i1i0

0111

f

3-valued logics – Bochvar

• Trying to avoid the paradoxes• i stands for ‘meaningless’

• {1} as the only designated value

• Any formula involving i is meaningless

• No tautologies, introducing ‘assertion’

10

ii

01

f 1i0

11i0

iiii

00i0

f

3-valued logics – Bochvar

1i0

11i0

iiii

01i1

f

• Łukasiewicz • Kleene• Bochvar

1i0

11i0

i11i

0111

f 1i0

11i0

i1ii

0111

f

1i0

11i0

iii0

0000

f 1i0

11i0

iii0

0000

f 1i0

11i0

iiii

00i0

f 1i0

11i0

iiii

00i0

f1i0

1111

i1ii

01i0

f 1i0

1111

i1ii

01i0

f

• What are the main differences?

Consequence Relations

• To simplify notation we work with – 1 two-place connective – 1 one-place connective

• A matrix for language L consist of – A non empty set, V of truth values– A non empty subset of V, D of the designated

values.– A set of interpretations for the connectives of

L (for every n-place connective c, an n-place function )

( * )

cf

Consequence Relations

• g is a substitution for a language L if it is a function from the set of variables into LWe write for the result of applying the substitution g to the variables in

• is consistent with respect to if there’s a formula so that

g

Consequence Relations

• A is a consequence of Σ with respect to the matrix M (Σ A) if for every assignment g, if for all B Σ then

• A is a tautology with respect to the matrix M if A

M

g(B) D g(A) D

M

Consequence Relations

• A consequence relation for a matrix M satisfies the following conditions:– –

• Matrix M is a characteristic matrix of an abstract sequence relation if it coincides with

M

, , ,M

M M M

if

if and then

M

Consequence Relations

• is a structural consequence relation if it satisfies the rule of uniform substitution:– If and g is a substitution then

• is uniform if it satisfies :– If and is

consistent, then

• If is a uniform structural consequence relation then has a characteristic matrix.

M g gM

M

M

, , ( { }) ( )Var Var

n-valued logics –

• Generalization of for n truth values

•

• truth functions :

nL

3L

| 0 11

kV k n

n

1

min{1,1 ( )}

min( , )

max( , )

x x

x y x y

x y x y

x y x y

n-valued logics - Post

• Truth values set = {0 … m-1}

•

• Designated values – {0 … n}

• Have a proof for completeness

nmT

min( , )

1 (mod )

x y x y

x x m

Dunn/Belnap 4-valued logic

•

• The two additional values represent uncertainty / inconsistency

• The truth degrees ordering is non-linear, two natural ways to order it are :

{ ,{ },{ },{ , }}V F T F T

Extension to first-order level

• We will use as an example

• We extend the language to contain:– Individual variables– Predicates– The quantifiers

• Given non empty set of individuals I– Constants a,b,c… to correlate with them

nL

,

Extension to first-order level

• An m-valued structure is an assignment of a truth-value to the atomic sentences in the Language.

• For in I and an n-place predicate P we give the value

• And for the quantifiers:

1... na a 1.. nPa a

min / |

max / |

x a x a I

x a x a I

Alternative - Supervaluations

• Given a interpretation v, v’ is a supervaluation of v if it is the same as v except where v(p)=i, in which case v’(p) is either 1 or 0.

• v’ represents one of the ways things could turn

3K

Alternative - Supervaluations

• We define in the following way:– Σ A iff for every interpretation v, every

supervaluation v’ that makes everytrue, makes A true.

• This definition reduces to truth preservation under all classical interpretations

• is just classical logic, although allowing truth-value gaps.

s

Bs

s

Fuzzy Logic (∞-valued logics)

• Fuzzy Logic is a type of MVL where the set of truth values is infinite

• Usually the set of all real values in [0,1] are used, where 1 is true.

Fuzzy Logic – Motivation

• Sorites paradox (the bald man / the heap)– (i) One grain of wheat does not make a heap– (ii) Adding one grain of wheat to something

that is not a heap, does not make it a heap

– Apply (ii) 9,999 times and you’ll get that 10,000 grains of wheat do not make a heap.

Fuzzy Logic – Motivation

• Can be displayed as a sequence of Modus Ponens:

0 0 1M M M

1 1 2M M M

1 1k k kM M M

2M

kM

. . .

Fuzzy Logic – Motivation

• is definitely true. is definitely false

• If there exist a unique i for which is true and is false, then is false.

• It is counterintuitive that one small hair will make the person bald.

iM2M kM

1iM 1i iM M

The continuum-valued logic - Ł

• Truth values – all real numbers in [0,1]

• Extension of for infinite truth-values

• Same truth functionsnL

1

min{1,1 ( )}

min( , )

max( , )

x x

x y x y

x y x y

x y x y

The continuum-valued logic - Ł

• Designated values– Are context dependent – We choose ε and will be

• Consequence relations– Σ A iff for all interpretation v, if

for all B Σ, then – Context independent : Σ A iff for all Σ A

D { | }x x

v(B) v(A)

The continuum-valued logic - Ł

• Modus Ponens is invalid when

• v(p)=½ → v(^p)=½ → v( )=½

• Let’s assume there are 3 balls (a,b)

– Red(a)=1, Small(a)=½

– Red(b)=½, Small(b)=½

– Which ball is better for Red & Small?

1 p p

Applications of MVL

• Linguistics – treating sentences with presuppositions (assumptions created by the sentence)

• Logic – understanding other systems of logic, merging modalities, and modeling truth-values gaps

• Philosophical – understanding the meaning of “truth” in paradoxes

Applications of MVL

• Hardware design – designing many valued switches (more than two voltage levels)

• Artificial Intelligence – using fuzzy logic for vague notions and commonsense reasoning.

• Mathematics – using the logical matrices as a technical tool for consistency proofs in the set theory

Some history

• Aristotle

• [~1920] Łukasiewicz

• [1952] Kleene

• [1975] Kripke suggested i should be lack of truth value and not a third one

• [1979] Priest created LP

• [1969] van Fraassen invented Supervaluations

3 & nL L

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Summary

• Last words and room for questions