Truth and Paradox. Leon Horsten. Summer School on Set Theory and Higher-Order Logic London, 1 6 August Truth and Paradox.

Transcription

1 Summer School on Set and Higher-Order Logic London, 1 6 August 2011

2 Structure of the Tutorial 1. Introduction: 2. Definition, of Truth 8.

3 Tarski-biconditionals The disquotational intuition: supposing φ is tantamount to supposing the truth of φ, and conversely. A plausible axiom scheme for truth: φ is true if and only if φ. This scheme has infinitely many instances Instances of this scheme are called Tarski-biconditionals. (after the logician Alfred Tarski)

4 Paradoxical sentences The liar sentence L: The truth teller: This sentence is not true. This sentence is true. L will be our paradigmatic test case but we should be aware that it is not the only paradoxical sentence

5 Semantic paradox The argument of the liar paradox: Proof. It is an instance of the Tarski biconditional scheme that L is true if and only if L. But L if and only if L is not true for this is what L says of itself. So L is true if and only if L is not true: a short truth table calculation convinces us that we have lapsed into inconsistency. How credible is this argument?

6 The framework For simplicity, we want a toy model for English. Language: The language of truth (L T ): the language of arithmetic plus a truth predicate T. Background theory: First-order Peano arithmetic (PA T ). [The truth predicate T is allowed in the induction scheme.]

7 The arithmetical background Why arithmetic? Coding PA can serve as a theory of syntax (via coding); PA allows the formation of self-referential sentences (via coding). Note: We will be sloppy about notation.

8 Defining truth Tarski showed us how to give a definition of truth for a formal language in purely logical and mathematical terms. material adequacy condition: a definition of truth for a language L should imply all the Tarski-biconditionals for sentences of L. Metawissenschaft

9 Defining first-order arithmetical truth in second-order arithmetic There is a first-order formula val + that defines the atomic arithmetical truths. Consider the condition Ψ(X, φ): φ atomic closed [X (φ) val + (φ)] ψ, λ : φ = ψ λ [X (φ) X (ψ) X (λ)]... Definition true(φ) X : λψ(x, λ) X (φ). The second-order arithmetical predicate true(x) satisfies Tarski s material adequacy condition.

10 Models for truth A theory of truth for a language L should describe a class of intended models for L. [This thesis was popular in the 1970s and 1980s.]

11 Universality We want a theory of truth for our language: We want a truth theory for English in English. Metalanguage = Objectlanguage

12 Definitions and universality Problem for the definitional approach: A truth definition for L (objectlanguage) can only be given in a language (metalanguage) L that is essentially richer than L: Theorem (Undefinability theorem I) There is no first-order arithmetical formula that defines first-order arithmetical truth. But perhaps a theory of truth for English as it now is can only be given in future English...

13 Models and universality A class of models for a language L is described in an essentially richer metalanguage (incompleteness theorem). Additional problem for the model-theoretic approach: the domain of a model is a set The domain of discourse of English does not form a set. But perhaps the techniques can be adapted to proper class domains...

14 Axioms for Truth Thesis A theory of truth for a language L should posit rules of inference and / or axioms for L. The problems of the definitional and the model-theoretic approach are not applicable. there is no immediately apparent obstacle to our universal ambitions Definition Let L T be L PA plus a primitive truth predicate T. Aim: Formulate a truth theory for L T in L T.

15 The role of models and definitions Models can have a strong heuristic force: they give pictures. We can give truth definitions for fragments of our language.

16 Soundness and completeness Desiderata for axiomatic truth : soundness truth theoretic completeness More specific desiderata will be discussed at the end of the lectures.

17 The naive theory of truth Recall the disquotational intuition. Take all the Tarski-biconditionals as your theory of truth. The formal theory NT : 1. PA T 2. T (φ) φ for all φ L T

18 Gödel s diagonal lemma Theorem (Gödel) For each formula φ(x) L T, there is a sentence λ L T such that PA T proves λ φ(λ). This sentence λ is self-referential.

19 Tarski s undefinability theorem Theorem (Undefinability theorem 2) No consistent extension S of PA T proves T (φ) φ for all φ L T. Proof. Use the diagonal lemma to produce a (liar) sentence λ: PA T λ T (λ). If the theory S in question indeed proves T (φ) φ for all φ L T, then in particular S proves T (λ) λ. Putting these equivalences together, we obtain a contradiction in S: S T (λ) T (λ). Consequence: NT is inconsistent.

20 The typed disquotational theory Tarski s diagnosis: the root of the disease lies in allowing the Tarski-biconditionals to regulate the truth-conditions of sentences that themselves contain the truth predicate (such as L). typed truth The disquotational theory DT : DT1 PA T ; DT2 T (φ) φ for all φ L PA. Tarski s strictures are respected.

21 Soundness Proposition DT has a nice model. Proof. Consider the model M =: N, {φ φ L PA N = φ}, i.e., the model in which as the extension of the truth predicate we take all arithmetical truths. An induction on the length of proofs in DT verifies that M = DT. Consequence: DT is arithmetically sound.

22 The Tarskian hierarchy 1 Conversation: A: It is true that 0 = 0. B: What you have just said is true. Proposition DT T (T (0 = 0))

23 The Tarskian hierarchy 2 The theory DT 1 : 1. PA T,T 1 ; 2. T (φ) φ for all φ L PA ; 3. T 1 (φ) φ for all φ L T. Theorem DT 1 has a nice model. Proposition DT 1 T 1 (T (0 = 0))

24 The Tarskian hierarchy 3 Extended conversation: A: It is true that 0 = 0. B: What you have just said is true. C: Yes, B, that is very true. the Tarskian hierarchy DT, DT 1, DT 2,... the notion of truth is irrevocably fragmented

25 Contextualist (Burge, Gaifman, Barwise & Perry, Glanzberg,... ) Thesis Truth is a uniform but indexical concept. Sentence L is not true 0. Sentence L is true 1. Because of the indexical shift in extension of the truth predicate between context 0 and context 1, this is not a contradiction.

26 The strengthened liar paradox (S) Sentence S is not true in any context. this is called a strengthened liar sentence it is not hard to figure out that there is no context in which S can be coherently evaluated.

27 The weakness of disquotationalism The intuition: truth distributes over the logical connectives. Proposition For all φ L PA : DT T (φ) T ( φ) Proof. Already propositional logic alone proves φ φ. Two restricted Tarski-biconditionals are T (φ) φ and T ( φ) φ. Combining these facts yields the result. Proposition (Tarski) DT φ L PA : T (φ) T ( φ)

28 The typed theory (Davidson...) The theory TC: TC1 PA T ; TC2 atomic φ L PA : T (φ) val + (φ); TC3 φ L PA : T ( φ) T (φ); TC4 φ, ψ L PA : T (φ ψ) (T (φ) T (ψ)); TC5 φ(x) L PA : T ( xφ(x)) xt (φ(x)). Theorem TC has nice models, and DT TC.

29 Substitution For a moment we have to be sticklers for notation... A sentence of the form xt (φ(x)) is really expressed along the following lines: There is a number x such that when the standard numeral for x is substituted for the variable x in φ(x), a true sentence results. Thus a substitution function (expressible in the language of arithmetic) appears in formulae such as TC5, but also in all the other axioms.

30 True and true of What if we do not have standard names for all elements of the language of discourse (R,... )? we work with a satisfaction relation ( true of ) and define the truth predicate in terms of it. The axiom TC5 then becomes (roughly): x, φ(y) : Sat(x, yφ(y)) zsat(z, φ(y)))

31 Deflationism and conservativeness Deflationism: The concept of truth does not play a substantial role in philosophical, mathematical, scientific debates. Definition A theory of truth S is arithmetically conservative over PA if for every sentence φ L PA, if S φ, then already PA φ. Proposition DT is arithmetically conservative over PA.

32 The non-conservativeness of truth 1 Theorem TC φ L PA : Bew PA (φ) T (φ), where Bew PA (...) is an arithmetical predicate that expresses provability in Peano arithmetic in a natural way. Proof. This is proved by an induction, inside TC, on the length of proofs. [Here we need that T is allowed in the induction scheme.] For the basis case, we have to prove that all the axioms of true. [For the subcase of mathematical induction we again need that T is allowed in the induction scheme.]

33 The non-conservativeness of truth 2 Corollary TC Bew PA (0 = 1). Proof. This follows from the previous theorem and the fact that DT TC by instantiating 0 = 1 for φ. So by Gödel s second incompleteness theorem, TC is not conservative over PA.

34 The non-conservativeness of truth 3 More precise information: Definition The second-order system ACA contains full second-order induction but only those instances of Comprehension X y : X (y) φ(y) where φ(y) contains no bound second-order quantifiers (and does not contain X free). Theorem The first-order arithmetical strength of TC is exactly that of the second-order system ACA.

35 Back to deflationism Thesis (deflationism) The concept of truth is not substantial but provides extra conceptual power. An axiomatic theory of truth must be conservative but noninterpretable.

36 A deflationist truth theory 1 Fischer proposed a minimally adequate truth theory PT Definition tot(φ(x)) y : T (φ(y) T ( φ(y))

37 A deflationist truth theory 2 PT-1 PA T without the induction axiom; PT-2 φ(x) L T : [tot(φ(x)) T (φ(0)) y(t (φ(y)) T (φ(y + 1))] xt (φ(x)); PT-3 atomic φ L PA : T (φ) val + (φ); PT-4 atomic φ L PA : T ( φ) val + (φ); PT-5 φ, ψ L PA : T (φ ψ) (T (φ) T (ψ)); PT-6 φ, ψ L PA : T ( (φ ψ)) (T ( φ) T ( ψ)); PT-7 φ(x) L PA : T ( xφ(x)) xt (φ(x)); PT-8 φ(x) L PA : T ( xφ(x)) xt ( φ(x)); PT-9 φ L PA : T ( φ) T (φ).

38 A deflationist truth theory 3 Observation: PT is still highly, but not as as TC. Theorem PT is conservative over PA but is not interpretable in PA. PT seems sufficient for capturing the technical use of the concept of truth that is made in mathematics (such as model theory).

39 truth Truth which are formulated in languages that contain truth predicates of different levels and which prove iterated truth ascriptions only if the hierarchy constraints are satisfied, are called typed of truth. There also exist truth systems which contain a single truth predicate but which do validate sentences of the form T (T (0 = 0)). These systems are called type-free of truth (reflexive, semantically closed ).

40 The strength of type-free disquotation Theorem (McGee) Any theory extending PA can be reaxiomatized by the axioms of PA and a set of Tarski-biconditionals. Proof. Consider an axiom ψ of S. Using the diagonal lemma, we can find a sentence such that λ (T (λ) ψ) is provable in PA. This equivalence is logically equivalent to: ψ (T (λ) λ). So ψ is PA-provably equivalent to the Tarski-biconditional T (λ) λ.

41 Positive disquotation (Halbach) The theory PUTB is the theory given by PA T and the set of all sentences x (T (φ(x)) φ(x)), where in the formula φ(x), T must not occur in the scope of an odd number of negation symbols in the formula φ. Theorem PUTB T (T (0 = 0)) But: PUTB φ L PA : T ( φ) T (φ)

42 Going partial (Kripke) Structure of the argument of the liar paradox: 1. L L 2. L 3. L So, Moral: Do not assert the law of excluded third.

43 Partial models The aim is build a model for the language L T in stages The arithmetical vocabulary is interpreted throughout as in the standard model N. The truth predicate T will be the only partially interpreted symbol: it will receive, at each ordinal stage, an extension E and an anti-extension A. Note: 1. E A = M = (E, A) 2. E A does not typically exhaust the domain, for otherwise T would be a total predicate.

44 Strong Kleene valuation For any atomic formula Fx 1...x n : 1. M = sk Fk 1...k n if the n-tuple (k 1,..., k n ) belongs to the extension of F ; 2. M = sk Fk 1...k n if the n-tuple (k 1,..., k n ) belongs to the anti-extension of F. For any formulae φ, ψ : 1. M = sk φ ψ if and only if M = sk φ and M = sk ψ; 2. M = sk (φ ψ) if and only if either M = sk φ or M = sk ψ (or both); 3. M = sk xφ if and only if for all n, M = sk φ(n/x); 4. M = sk xφ if and only if for at least one n, M = sk φ(n/x); 5. M = sk φ if and only if M = sk φ.

45 A chain of partial models 1 M 0 = (E 0, A 0 ) =: (, ) ; E α+1 =: {φ L T M α = sk φ} and A α+1 =: {φ L T M α = sk φ} ; For λ limit ordinal, we set: E λ =: E κ, κ<λ A λ =: A κ. κ<λ

46 A chain of partial models 2 This inductive definition gives rise to a transfinite chain of partial models: (E 0, A 0 ), (E 1, A 1 ),..., (E ω, A ω ),... Theorem (monotonicity) For any two partial models (E a, A a ), (E b, A b ), if E a E b and A a A b, then {φ (E a, A a ) = sk φ} {φ (E b, A b ) = sk φ}.

47 The minimal fixed point model A consequence of this is that: Corollary For all α, β with α < β, we have: {φ (E α, A α ) = sk φ} {φ (E β, A β ) = sk φ}. Proposition For some ordinal ρ, E ρ = E ρ+1 and A ρ = A ρ+1. This ordinal ρ is called the (Strong Kleene) minimal fixed point, and (E ρ, A ρ ) is called the minimal fixed point model. Theorem (Kripke) The minimal Strong Kleene fixed point is a Π 1 1-complete set.

48 Almost having it all The minimal fixed point almost makes the unrestricted Tarski-biconditionals true: Theorem For all sentences φ in L T : M ρ = sk φ M ρ = sk T (φ). Proof. First, suppose M ρ = sk φ. Then by the definition of the sequence of partial models, M ρ+1 = sk T (φ). But since M ρ is a fixed point, we have M ρ+1 = M ρ. So M ρ = sk T (φ). Second, suppose M ρ = sk T (φ). Then there must be an ordinal α < ρ such that M α = sk φ. And therefore, by monotonicity, M ρ = sk φ.

49 Paradoxical sentences Definition Closing off a partial model means letting its anti-extension spill over into all of the complement of its extension. Theorem M ρ sk T (L) and M ρ sk T (L). Proof. By closing off the model M ρ we obtain a classical model M c ρ. Suppose that M ρ = sk T (L), and thereby M ρ = sk L. Then by monotonicity, also M c ρ = sk T (L) and M c ρ = sk L. But since M c ρ is just a classical model, the diagonal lemma holds in it. So we have M c ρ = sk L T (L). But putting these three facts together gives us a contradiction. So we deny our supposition and conclude that M ρ sk T (L).

50 Other valuation schemes So far we have been using the Strong Kleene valuation scheme... Weak Kleene: if a component is gappy, then the whole is gappy. (Feferman) Paraconsistent: Truth gluts instead of truth gaps (Priest)... Formally, it does not make a lot of difference...

51 Supervaluation In the supervaluation approach, a formula φ L T is regarded as true in a partial model M = (E, A) if and only if φ is true in all total (or classical) models M c = N, C for which the interpretation C of the truth predicate is such that E C and A N\C. Similarly, we say that a formula φ L T is regarded as false in a partial model M = (E, A) if and only if φ is false in all total (or classical) models M c = N, C for which the interpretation C of the truth predicate is such that E C and A N\C.

52 Properties of supervaluation fixed points The laws of classical logic are supervaluation-true in the minimal fixed point of the supervaluation scheme. The supervaluation fixed point is not. Theorem The supervaluation fixed point is a complete Π 1 1 set.

53 Axiomatising Strong Kleene Fixed Points The theory KF : KF1 atomic φ L PA : T (φ) val + (φ); KF2 atomic φ L PA : T ( φ) val (φ); KF3 φ L T : T ( φ) T (φ); KF4 φ, ψ L T : T (φ ψ) (T (φ) T (ψ)); KF5 φ, ψ L T : T ( (φ ψ)) (T ( φ) T ( ψ)); KF6 φ (x) L T : T ( xφ (x)) yt (φ (y)); KF7 φ (x) L T : T ( xφ (x)) yt ( φ (y)); KF8 φ L T : T (T (φ)) T (φ); KF9 φ L T : T ( T (φ)) T ( φ); KF10 φ L T : (T (φ) T ( φ)).

54 Inner logic and outer logic Theorem KF has nice models. Proof. The closed off minimal fixed point model. Definition The inner logic of KF is the collection of sentences φ L T such that KF T (φ). Proposition KF L T (L), where L is the liar sentence. So the inner logic of KF does not coincide with the external logic of KF.

55 Ramified Analysis Let us add new second-order variables X 1, Y 1,... to the language of second-order arithmetic. Add a new comprehension principle X 1 y : X 1 (y) φ(y) where φ(y) contains no bound second-order quantifiers of the new kind (and does not contain X 1 free). The resulting system is called ACA 1.

56 The strength of KF We can go on, and construct ACA n,..., ACA <ω... Definition ɛ 0 is the least upper bound of {ω, ω ω, ω ωω,...}. Theorem (Feferman) The first order arithmetical theorems of KF are exactly those of ACA ɛ0.

57 Truth and Classes Definition x y y L T T (y(x)) Then KF can be seen as a theory of definable or predicative classes (Feferman). KF as a way of ascending from PA to predicative analysis.

58 A way out? What is the source of the divergence between inner and outer logic? Motivation of Kripke s theory = partial Logic of KF = classical Mismatch Solution: Formalise Kripke s theory in partial logic!

59 Logic and mathematics Restricted conditionalisation: T (φ) T ( φ) φ ψ Rule of induction: φ (Hyp). ψ (Hyp) φ(0) φ(x) (Hyp). φ(s(x)) (Hyp) xφ(x) Here φ(x) ranges over all formulae of L T.

60 The system PKF PKF1 PKF2 PKF3 PKF4 val + (t 1 = t 2 ) T (t 1 = t 2 ) T (φ) T (ψ) T (φ ψ) T (φ) T (ψ) T (φ ψ) xt (φ(x)) T ( xφ(x)) T (t 1 = t 2 ) val + (t 1 = t 2 ) T (φ ψ) T (φ) T (ψ) T (φ ψ) T (φ) T (ψ) T ( xφ(x)) xt (φ(x))

61 The system PKF Ctd PKF5 PKF6 PKF7 xt (φ(x)) T ( xφ(x)) T (φ) T (T (φ)) T (φ) T ( φ) T ( xφ(x)) xt (φ(x)) T (T (φ)) T (φ) T ( φ) T (φ)

62 Properties of PKF Theorem Inner logic of PKF = Outer logic of PKF Theorem PKF holds in all fixed point models of the Strong Kleene construction. Theorem The first-order arithmetical theorems of PKF are those of ACA ω ω.

63 A chain of classical models M 0 =: N, M α+1 =: N, {φ L T M α = φ}. For λ a limit ordinal: M λ =: N, {φ L T β γ : (γ β γ < λ) M γ = φ}.

64 Stable truth Stable truth: A sentence φ L T is said to be stably true if at some ordinal stage α, φ enters in the extension of the truth predicate of M α and stays in the extension of the truth predicate in all later models. Stable falsehood: A sentence φ L T is said to be stably false if at some ordinal stage α, φ is outside the extension of the truth predicate of M α and stays out forever thereafter.

65 Nearly stable truth Nearly stable truth: A sentence φ L T is said to be nearly stably true if for every stage α after some stage β, there is a natural number n such that for all natural numbers m n, φ is in the extension of the truth predicate of M α+m. Nearly stable falsehood: A sentence φ L T is said to be nearly stably false if for every stage α after some stage β, there is a natural number n such that for all natural numbers m n, φ is outside the extension of the truth predicate of M α+m.

66 Properties of the (nearly) stable truths The liar sentence is neither (nearly) stably true nor (nearly) stably false. Proposition The chain of revision models is eventually periodic. Theorem The (nearly) stable truths form a set that is more complicated than a complete Π 1 1 set.

67 The Friedman-Sheard theory 1 The theory FS: FS1 PA T ; FS2 atomic φ L PA : T (φ) val + (φ); FS3 φ L T : T ( φ) T (φ); FS4 φ, ψ L T : T (φ ψ) T (φ) T (ψ); FS5 φ(x) L T : T ( xφ(x)) xt (φ(x)). NEC From a proof of φ, infer T (φ); CNEC From a proof of T (φ), infer φ.

68 The Friedman-Sheard theory 2 Proposition Inner logic of FS = outer logic of FS Proposition FS is not stably true. Proof. FS3 is not stably true: both the liar sentence and its negation are false at all limit ordinals Proposition FS is nearly stably true.

69 The strength of FS FS only proves finite truth-iterations. Theorem The first-order arithmetical consequences of FS are exactly those of ACA ω. Corollary FS is arithmetically sound.

70 Omega-inconsistency Even though FS is consistent and indeed even arithmetically sound, it is in some sense almost inconsistent : Definition An arithmetical theory T is ω-inconsistent if for some formula φ(x), the theory T proves xφ(x) while at the same time for every n N, T proves φ(n) Theorem (McGee) For some formula φ (x) L T : FS xφ (x) and FS φ (n) for all n N.

71 A fork in the road FS-like DT TC KF-like

72 Desiderata for axiomatic truth Coherence Tarski-biconditionals ity Sustaining ordinary reasoning Strength Capturing a picture Fulfillment of these norms is a matter of degree.

73 Cantini, A. Logical Frameworks for Truth and Abstraction. North-Holland, Feferman, S. Reflecting on Incompleteness. Journal of Symbolic Logic 56(1991), p Gupta, A. & Belnap, N. of Truth. MIT Press, Halbach, V. Axiomatic of Truth. Cambridge University Press, Horsten L. The Tarskian Turn. Deflationism and axiomatic truth. MIT Press, McGee, V. Truth, Vagueness and Paradox. An essay on the logic of truth. Hackett, Visser, A. Semantics and the Liar Paradox. In: D. Gabbay et al (eds) Handbook of Philosophical Logic. Volume 4. Reidel, 1984, p