on the meaning of truth degrees

BackgroundFormalizing truth degrees in PALTr

counterexample of the formalized truth degree theoryConclusion

On the meaning of truth degrees

Shunsuke Yatabe

Research Center for Verification and Semantics,National Institute of Advanced Industrial Science and Technology,

Japan

February 20, 2010

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Abstract

I Primary objective: to introduce an a conflict of semanticaccount of truth and axiomatic account of truth

I Secondary objective: to analyze the truth conception infuzzy logics by formalizing “truth degrees”

I Motivation: try to explain how truth degrees relate to truthconception

I Methodology: to formalize truth degree theory in axiomatictruth theory PAŁTr2.

I Discussion: Since PAŁTr2 is ω-inconsistent, the formalizedtruth degree theory fails.

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On the meaning of fuzzy truth valuesAxiomatic approachTransparency of the truth conception

The framework logic

Łukasiewicz infinite-valued predicate logic ∀Ł is defined as follows:

(1) Truth values are real numbers in [0, 1],(2) ‖ϕ0 → ϕ1‖ = min{1, 1 − ‖ϕ0‖ + ‖ϕ1‖}, ‖⊥‖ = 0,

I ¬A ≡ A → ⊥, etc.

(3) ‖(∀x)ϕ(x)‖ = inf{‖ϕ(a)‖M : a ∈ |M|}.

∀Ł is a sublogic of classical logic (i.e. ∀Ł ` ϕ implies CL` ϕ).

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Motivation: semantic account of truth in fuzzy logics

I Often said: [0, 1] are truth values,I due to historical reason (e.g. Łukasiewicz, Zadeh)

I We can define truth degreesI we can construct degrees of all sentences in any algebra from

a viewpoint of metatheory:for any sentence A, B,

A ≤ B if and only if ‖A → B‖ = 1

I we call this ordering “truth degrees”, and often think that theyrepresent “degrees of truthhood” of sentences.

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Motivation: fuzzy truth values and algebra (1)

I Not all semantic objects are called “truth values”:I example: sets of possible worlds, situations, etc.

I Many fuzzy logics are characterized by their algebras, but it isnot trivial to say “such algebraic values are truth values ofsuch fuzzy logics”,

I Analogy: intutionistic logic caseI it is not complete for Tarskian semantics with truth values{0, 1}: two truth values are not appropriate for interprettingintutionistic logic.

I To say “the Heyting algebra is a truth value of intuitionisticlogic” is controversial: no constructivist agrees this.

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Motivation: fuzzy truth values and algebra (2)

I The problem: [0, 1] are not enough to interpret many fuzzylogics.

I some fuzzy predicate logics (as BL∀) are not complete for[0, 1],

I What are truth values of such logics?

I Sticking around “fuzzy truth values” comes with a heavy price:we can’t give a unified account to explain the meaning ofall fuzzy logics.

I The first proposal: to have the unified account has the firstpriority.

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Motivation the relationship between “truth degrees” and truth

I Instead of asking the meaning of truth values [0, 1] ....I Question: what is a relationship between the conception of

truth and the so called “truth degrees”?I if [0,1] are truth values, it is trivial,I if not: the ordering of truth degrees in fuzzy logics are

regarded as an abstraction of the order relation in thealgebraic semantics (e.g. Paoli).

I but we never call the chain in Heyting algebra “truth degrees”....I truth degrees should be about truth, but relationship between

algebraic value and truth is not trivial.I Asking the meaning of truth degrees is asking the meaning of

“truth values” of fuzzy logic in a roundabout way.

I The next goal:I to find a framework to formalize truth degree theory without

mentioning truth values,I to formalize truth degree theory within it.

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T-scheme and disquotation

I Alternative framework: axiomatic truth theoryI First we introduce its motivation.

Tarskian definition of truth:

Tr(dϕe) ≡ ϕ

for any formula ϕ.I sentence “Snow is white.” is true iff snow is white.I “disquotation view of truth” (Quine, etc.):

The role of truth predicate seems to “disquote” quotedsentences dϕe (then we get ϕ).

I According to them, truth does not have a significant role insemantics.

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The liar paradox

However, we can define the liar sentence:I “This sentence is false” (if the language has an indexical)I L ≡ ¬Tr(dLe)

I in case truth predicate is contained in the language, or it isdefinable in that theory,

I we can define λ in arithmetic by diagonalization argument.

L ∨ ¬L[v : L]

[v : L]L ↔ ¬LL → ¬L¬L

⊥[w : ¬L]

[w : ¬L]L ↔ ¬L¬L → L

L⊥

⊥ ∨−P LC

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Two ways out

I The solution 1: to sustain classical logic:I basic strategy:

I to restrict the domain of truth predicate (to exclude the liarsentence),

I axiomatic truth theory case:I to restrict T-scheme not to prove L, e.g. (McGee [M85])

Tr(dϕe) 6→ ϕ

Such restrictions prevents that the theory implies the liarsentence as a theorem.

I The solution 2: to sustain totality (and full T-scheme) of truthpredicate

I abandon classical logic

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Our framework PAŁTr2

I The framework to analyze truth conception of ∀Ł: PAŁTr2[HPS00] over ∀Ł

I whose axioms are all axioms of classical PA,I the induction scheme for formulae possibly containing the truth

predicate Tr andI T-schemata for a total truth predicate Tr(x)

ϕ ≡ Tr(dϕe)

where dϕe is the Godel code of ϕ.I The total truth predicate is not contradictory in PAŁTr2.

I The liar sentence, L ≡ ¬Tr(dLe), dose not imply acontradiction in ∀Ł: ‖L‖ = 0.5,

I We can have a semantically closed language of arithmetic in∀Ł.

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Transparency of the truth conception in PAŁTr2

I Since the total truth predicate exists, their truth conceptionsseem to be transparent,

I i.e. no theoretical restriction on the domain of Tr (as ”Tr can’tbe applied to the liar sentence) are made

I philosophically, they are successors of disquotational view oftruth.

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Formalization: “truth degrees” in terms of axiomatic truththeory

I We define ≤ as follows (call this “degree theoretic ordering”):

dϕe ≤ dψe ≡ ϕ → ψ

I We define ≺ as follows (call this “ordering of truthhood”):I define an ordering ≺⊆ ω × ω as follows:

dϕe ≺ dψe ≡ Tr(dϕe) → Tr(dψe)I ≺ is defined by the conditionals of the form “a truthhood of

some formula implies a truthhood of another formula”.I Therefore these conditionals definitely represent “degrees of

truthhood” in the sense of truth theory [Fl08].I Truth degree theory says two orderings are identical: for any

formula ϕ, ψ,dϕe ≤ dψe ≡ dϕe ≺ dψe

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Formzlization: the formalized truth degree theory in PAŁTr2

Before we answer the question, we formalize the truth degreetheory in PAŁTr2.I We define ≤, ≺ between Godel codes of formulae of PAŁTr2.

I (∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ (Tr(x→y))],I (∀x, y)(Form(x)&Form(y) → [x ≺ y ≡ (Tr(x) → Tr(y))],

I The formalized truth degree theory identity degreetheoretic ordering (≤) with degrees of truthhood (≺) :

(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y])

I The ordering need not to be linearly ordered: e.g. Paoli’s“really fuzzy” truth degrees.

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A (pathological) counter example

I We fix PAŁTr2 as a object theory and metatheory.

I PAŁTr2 shows that ≤ and ≺ are not identical:

(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y]) → ⊥

I if Tr(x→y) ≡ (Tr(x) → Tr(y)), then mathematical inductionimplies the contradictory sentences [HPS00].

In this sense, the assumption of truth degree theory need nothold.

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Remarks

I For any formula ϕ, ψ, PAŁTr2 proves the following:

dϕe ≤ dψe ≡ dϕe ≺ dψe

I However, PAŁTr2 proves, the following formalizedcommutativity implies a contradiction:

(∀x, y)(Form(x)&Form(y) → [Tr(x→y) ≡ (Tr(x) → Tr(y))])

This fails when x or y is a non-standard natural number(ω-inconsistency! [R93]).

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An objection and replies (1)

I Objection: HPS paradox merely shows that PAŁTr2 is notsuitable framework to analyze the conception of “Truthdegrees”.

I ω-inconsistency is a crucial crux [Fl08],I This is not a failure of truth degree theory, but a failure of

axiomatic theory (hajek).

I Reply:

I Defensive: Even though PAŁTr2 are pathological, it providesa precise distinction of two concepts (as constructivemathematics gives a distinction between propositions whichare equivalent in classical logic).

I Offensive: Since T-scheme is the key concept of truth, andinduction is essential to arithmetic, we must think theconsequence of PAŁTr2 seriously.

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An objection and replies (2)

I Objection: What is a meaning of non-standard elements ofForm?

I Reply:I In PAŁTr2, we can represent an infinite operation on a formula

by some formula: taking a sup ofA, ¬A → A, ¬A → (¬A → A), · · · .

I PAŁTr2 is a theory based on an extension of PA to representinfinite processes in PAŁTr2 itself.

I we can define an arithmetical function which corresponds aninfinite operation on codes,

I it is interpreted to the real operation on formulae by using Tr,I this enables to treat an infinite process as an object in PAŁTr2,I non-standard numbers and non-standard elements of Form

represent such infinite processes.

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Conclusion

I We try to formalize truth degree theory without mentioningtruth values.

I We can define degree theoretic ordering ≤, but it is not trivialthat how such degree relate truth.

I If we want to formalize “truth degree”, we needed a truthpredicate and truth theoretic machinery.

I Truth degree theory can be formalized as supposing thatdegree theoretic ordering ≤ is isomorphic to degrees oftruthhood ≺.

I However some truth theory provides its counterexamplebecause of ω-inconsistency.

I This means that, sometimes semantic anaysis and axiomaticanalysis have differing opinions.

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Reference

Hartry Field. “Saving Truth From Paradox” Oxford (2008)

Petr Hajek, Jeff B. Paris, John C. Shepherdson. “ The Liar Paradoxand Fuzzy Logic” Journal of Symbolic Logic, 65(1) (2000) 339-346.

Hannes Leitgeb. “Theories of truth which have no standard models”Studia Logica, 68 (2001) 69-87.

Vann McGee. “How truthlike can a predicate be? A negative result”Journal of Philosophical Logic, 17 (1985): 399-410.

Robin Milner, Mads Tofte. “Co-induction in relational semantics”Theoretical computer science 87 (1991) 209-220.

Greg Restall “Arithmetic and Truth in Łukasiewicz’s Infinitely ValuedLogic” Logique et Analyse 36 (1993) 25-38.

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on the meaning of truth degrees

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