On the meaning of truth degrees

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応用哲学会・冬の研究大会・特別ワークショップ 「哲学における非古典論理の役割」における発表(2010年2月20日)

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On the meaning of truth degrees

  1. 1. <p><strong>Slide 1: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion 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 1 / 20 </p><p><strong>Slide 2: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Abstract Primary objective: to introduce an a conflict of semantic account of truth and axiomatic account of truth Secondary objective: to analyze the truth conception in fuzzy logics by formalizing “truth degrees” Motivation: try to explain how truth degrees relate to truth conception Methodology: to formalize truth degree theory in axiomatic truth theory PAŁTr2 -&gt; Discussion: Since PAŁTr2 is ω-inconsistent, the formalized truth degree theory fails-&gt; 2 / 20 </p><p><strong>Slide 3: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion 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, ¬A ≡ A → ⊥, etc-&gt; (3) (∀x)ϕ(x) = inf{ ϕ(a) M : a ∈ |M|}-&gt; ∀Ł is a sublogic of classical logic (i-&gt;e-&gt; ∀Ł ϕ implies CL ϕ)-&gt; 3 / 20 </p><p><strong>Slide 4: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Motivation: semantic account of truth in fuzzy logics Often said: [0, 1] are truth values, due to historical reason (e-&gt;g-&gt; Łukasiewicz, Zadeh) We can define truth degrees 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 we call this ordering “truth degrees”, and often think that they represent “degrees of truthhood” of sentences-&gt; 4 / 20 </p><p><strong>Slide 5: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Motivation: fuzzy truth values and algebra (1) Not all semantic objects are called “truth values”: example: sets of possible worlds, situations, etc-&gt; Many fuzzy logics are characterized by their algebras, but it is not trivial to say “such algebraic values are truth values of such fuzzy logics”, Analogy: intutionistic logic case it is not complete for Tarskian semantics with truth values {0, 1}: two truth values are not appropriate for interpretting intutionistic logic-&gt; To say “the Heyting algebra is a truth value of intuitionistic logic” is controversial: no constructivist agrees this-&gt; 5 / 20 </p><p><strong>Slide 6: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Motivation: fuzzy truth values and algebra (2) The problem: [0, 1] are not enough to interpret many fuzzy logics-&gt; some fuzzy predicate logics (as BL∀) are not complete for [0, 1], What are truth values of such logics? Sticking around “fuzzy truth values” comes with a heavy price: we can’t give a unified account to explain the meaning of all fuzzy logics-&gt; The first proposal: to have the unified account has the first priority-&gt; 6 / 20 </p><p><strong>Slide 7: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Motivation the relationship between “truth degrees” and truth Instead of asking the meaning of truth values [0, 1] -&gt;-&gt;-&gt;-&gt; Question: what is a relationship between the conception of truth and the so called “truth degrees”? if [0,1] are truth values, it is trivial, if not: the ordering of truth degrees in fuzzy logics are regarded as an abstraction of the order relation in the algebraic semantics (e-&gt;g-&gt; Paoli)-&gt; but we never call the chain in Heyting algebra “truth degrees”-&gt;-&gt;-&gt;-&gt; truth degrees should be about truth, but relationship between algebraic value and truth is not trivial-&gt; Asking the meaning of truth degrees is asking the meaning of “truth values” of fuzzy logic in a roundabout way-&gt; The next goal: to find a framework to formalize truth degree theory without mentioning truth values, to formalize truth degree theory within it-&gt; 7 / 20 </p><p><strong>Slide 8: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion T-scheme and disquotation Alternative framework: axiomatic truth theory First we introduce its motivation-&gt; Tarskian definition of truth: Tr( ϕ ) ≡ ϕ for any formula ϕ-&gt; sentence “Snow is white-&gt;” is true iff snow is white-&gt; “disquotation view of truth” (Quine, etc-&gt;): The role of truth predicate seems to “disquote” quoted sentences ϕ (then we get ϕ)-&gt; According to them, truth does not have a significant role in semantics-&gt; 8 / 20 </p><p><strong>Slide 9: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion The liar paradox However, we can define the liar sentence: “This sentence is false” (if the language has an indexical) L ≡ ¬Tr( L ) in case truth predicate is contained in the language, or it is definable in that theory, we can define λ in arithmetic by diagonalization argument-&gt; L ↔ ¬L L ↔ ¬L [v : L] L → ¬L [w : ¬L] ¬L → L [v : L] ¬L [w : ¬L] L L ∨ ¬L ⊥ ⊥ − ⊥ ∨ LC P 9 / 20 </p><p><strong>Slide 10: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Two ways out The solution 1: to sustain classical logic: basic strategy: to restrict the domain of truth predicate (to exclude the liar sentence), axiomatic truth theory case: to restrict T-scheme not to prove L, e-&gt;g-&gt; (McGee [M85]) Tr( ϕ ) → ϕ Such restrictions prevents that the theory implies the liar sentence as a theorem-&gt; The solution 2: to sustain totality (and full T-scheme) of truth predicate abandon classical logic 10 / 20 </p><p><strong>Slide 11: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Our framework PAŁTr2 The framework to analyze truth conception of ∀Ł: PAŁTr2 [HPS00] over ∀Ł whose axioms are all axioms of classical PA, the induction scheme for formulae possibly containing the truth predicate Tr and T-schemata for a total truth predicate Tr(x) ϕ ≡ Tr( ϕ ) where ϕ is the Godel code of ϕ-&gt; ¨ The total truth predicate is not contradictory in PAŁTr2 -&gt; The liar sentence, L ≡ ¬Tr( L ), dose not imply a contradiction in ∀Ł: L = 0-&gt;5, We can have a semantically closed language of arithmetic in ∀Ł-&gt; 11 / 20 </p><p><strong>Slide 12: </strong> Background On the meaning of fuzzy truth values Formalizing truth degrees in PALTr Axiomatic approach counterexample of the formalized truth degree theory Transparency of the truth conception Conclusion Transparency of the truth conception in PAŁTr2 Since the total truth predicate exists, their truth conceptions seem to be transparent, i-&gt;e-&gt; no theoretical restriction on the domain of Tr (as ”Tr can’t be applied to the liar sentence) are made philosophically, they are successors of disquotational view of truth-&gt; 12 / 20 </p><p><strong>Slide 13: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Formalization: “truth degrees” in terms of axiomatic truth theory We define ≤ as follows (call this “degree theoretic ordering”): ϕ ≤ ψ ≡ϕ→ψ We define ≺ as follows (call this “ordering of truthhood”): define an ordering ≺⊆ ω × ω as follows: ϕ ≺ ψ ≡ Tr( ϕ ) → Tr( ψ ) ≺ is defined by the conditionals of the form “a truthhood of some formula implies a truthhood of another formula”-&gt; Therefore these conditionals definitely represent “degrees of truthhood” in the sense of truth theory [Fl08]-&gt; Truth degree theory says two orderings are identical: for any formula ϕ, ψ, ϕ ≤ ψ ≡ ϕ ≺ ψ 13 / 20 </p><p><strong>Slide 14: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Formzlization: the formalized truth degree theory in PAŁTr2 Before we answer the question, we formalize the truth degree theory in PAŁTr2 -&gt; We define ≤, ≺ between Godel codes of formulae of PAŁTr2 -&gt; ¨ (∀x, y)(Form(x)&amp;Form(y) → [x ≤ y ≡ (Tr(x→y))], ˙ (∀x, y)(Form(x)&amp;Form(y) → [x ≺ y ≡ (Tr(x) → Tr(y))], The formalized truth degree theory identity degree theoretic ordering (≤) with degrees of truthhood (≺) : (∀x, y)(Form(x)&amp;Form(y) → [x ≤ y ≡ x ≺ y]) The ordering need not to be linearly ordered: e-&gt;g-&gt; Paoli’s “really fuzzy” truth degrees-&gt; 14 / 20 </p><p><strong>Slide 15: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion A (pathological) counter example We fix PAŁTr2 as a object theory and metatheory-&gt; PAŁTr2 shows that ≤ and ≺ are not identical: (∀x, y)(Form(x)&amp;Form(y) → [x ≤ y ≡ x ≺ y]) → ⊥ if Tr(x→y) ≡ (Tr(x) → Tr(y)), then mathematical induction ˙ implies the contradictory sentences [HPS00]-&gt; In this sense, the assumption of truth degree theory need not hold-&gt; 15 / 20 </p><p><strong>Slide 16: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Remarks For any formula ϕ, ψ, PAŁTr2 proves the following: ϕ ≤ ψ ≡ ϕ ≺ ψ However, PAŁTr2 proves, the following formalized commutativity implies a contradiction: (∀x, y)(Form(x)&amp;Form(y) → [Tr(x→y) ≡ (Tr(x) → Tr(y))]) ˙ This fails when x or y is a non-standard natural number (ω-inconsistency! [R93])-&gt; 16 / 20 </p><p><strong>Slide 17: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion An objection and replies (1) Objection: HPS paradox merely shows that PAŁTr2 is not suitable framework to analyze the conception of “Truth degrees”-&gt; ω-inconsistency is a crucial crux [Fl08], This is not a failure of truth degree theory, but a failure of axiomatic theory (hajek)-&gt; Reply: Defensive: Even though PAŁTr2 are pathological, it provides a precise distinction of two concepts (as constructive mathematics gives a distinction between propositions which are equivalent in classical logic)-&gt; Offensive: Since T-scheme is the key concept of truth, and induction is essential to arithmetic, we must think the consequence of PAŁTr2 seriously-&gt; 17 / 20 </p><p><strong>Slide 18: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion An objection and replies (2) Objection: What is a meaning of non-standard elements of Form? Reply: In PAŁTr2 , we can represent an infinite operation on a formula by some formula: taking a sup of A, ¬A → A, ¬A → (¬A → A), · · · -&gt; PAŁTr2 is a theory based on an extension of PA to represent infinite processes in PAŁTr2 itself-&gt; we can define an arithmetical function which corresponds an infinite operation on codes, it is interpreted to the real operation on formulae by using Tr, this enables to treat an infinite process as an object in PAŁTr2 , non-standard numbers and non-standard elements of Form represent such infinite processes-&gt; 18 / 20 </p><p><strong>Slide 19: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Conclusion We try to formalize truth degree theory without mentioning truth values-&gt; We can define degree theoretic ordering ≤, but it is not trivial that how such degree relate truth-&gt; If we want to formalize “truth degree”, we needed a truth predicate and truth theoretic machinery-&gt; Truth degree theory can be formalized as supposing that degree theoretic ordering ≤ is isomorphic to degrees of truthhood ≺-&gt; However some truth theory provides its counterexample because of ω-inconsistency-&gt; This means that, sometimes semantic anaysis and axiomatic analysis have differing opinions-&gt; 19 / 20 </p><p><strong>Slide 20: </strong> Background Formalizing truth degrees in PALTr counterexample of the formalized truth degree theory Conclusion Reference Hartry Field-&gt; “Saving Truth From Paradox” Oxford (2008) ´ Petr Hajek, Jeff B-&gt; Paris, John C-&gt; Shepherdson-&gt; “ The Liar Paradox and Fuzzy Logic” Journal of Symbolic Logic, 65(1) (2000) 339-346-&gt; Hannes Leitgeb-&gt; “Theories of truth which have no standard models” Studia Logica, 68 (2001) 69-87-&gt; Vann McGee-&gt; “How truthlike can a predicate be? A negative result” Journal of Philosophical Logic, 17 (1985): 399-410-&gt; Robin Milner, Mads Tofte-&gt; “Co-induction in relational semantics” Theoretical computer science 87 (1991) 209-220-&gt; Greg Restall “Arithmetic and Truth in Łukasiewicz’s Infinitely Valued Logic” Logique et Analyse 36 (1993) 25-38-&gt; 20 / 20 </p>

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