Axiomatic Theories of Truth

Transcription

1 Axiomatic Theories of Truth Graham Leigh University of Leeds LC 8, 8th July 28 Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 1 / 15

2 Introduction Formalising Truth Formalising Truth Definition (Language of truth) We wor in L T the language of Peano Arithmetic augmented with an additional predicate symbol T. Let PA T denote PA formulated in the language L T. The intuition is that T (x) denotes that x is (the Gödel number of) a true L T sentence. Let. provide a Gödel numbering of L T. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 2 / 15

3 Introduction Formalising Truth Formalising Truth Definition (Language of truth) We wor in L T the language of Peano Arithmetic augmented with an additional predicate symbol T. Let PA T denote PA formulated in the language L T. The intuition is that T (x) denotes that x is (the Gödel number of) a true L T sentence. Let. provide a Gödel numbering of L T. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 2 / 15

4 Introduction Formalising Truth Choices for truth Of course, by Tarsi s Theorem the ideal axiom of truth, T A A for all sentences A, is inconsistent with PA T. However, there are ways in which we can overcome this inconsistency. 1 Restrict the language so as to stop self-reference. For example allow T A A for L PA. 2 Replace T A A with weaer, consistent, axioms. We will consider case 2. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 3 / 15

5 Introduction Formalising Truth Choices for truth Of course, by Tarsi s Theorem the ideal axiom of truth, T A A for all sentences A, is inconsistent with PA T. However, there are ways in which we can overcome this inconsistency. 1 Restrict the language so as to stop self-reference. For example allow T A A for L PA. 2 Replace T A A with weaer, consistent, axioms. We will consider case 2. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 3 / 15

6 Introduction Formalising Truth Choices for truth Of course, by Tarsi s Theorem the ideal axiom of truth, T A A for all sentences A, is inconsistent with PA T. However, there are ways in which we can overcome this inconsistency. 1 Restrict the language so as to stop self-reference. For example allow T A A for L PA. 2 Replace T A A with weaer, consistent, axioms. We will consider case 2. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 3 / 15

7 Introduction Formalising Truth Choices for truth Of course, by Tarsi s Theorem the ideal axiom of truth, T A A for all sentences A, is inconsistent with PA T. However, there are ways in which we can overcome this inconsistency. 1 Restrict the language so as to stop self-reference. For example allow T A A for L PA. 2 Replace T A A with weaer, consistent, axioms. We will consider case 2. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 3 / 15

8 Introduction A Base for truth A Base for truth Let Base T be the theory comprising of PA T and the following axioms. 1 (T A B T A ) T B. 2 T (ucl B ) for all tautologies B. 3 T A if A is a true primitive recursive atomic sentence. where ucl(a) denotes the (Gödel number of the) universal closure of A. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 4 / 15

9 Axioms of truth Introduction Axioms for truth. Possible axioms, schema, and rules of inference we consider are A T A, (T A T A ), A/T A, T A A, T A T A, T A /A, T A T T A, n T Aṅ T x Ax, A/ T A, T T A T A, T x Ax n T Aṅ, T A / A. These axioms were considered by Harvey Freidman and Michael Sheard in An axiomatic approach to self-referential truth [2]. They classified the above axioms and rules into nine maximally consistent sets. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 5 / 15

10 Axioms of truth Introduction Axioms for truth. Possible axioms, schema, and rules of inference we consider are A T A, (T A T A ), A/T A, T A A, T A T A, T A /A, T A T T A, n T Aṅ T x Ax, A/ T A, T T A T A, T x Ax n T Aṅ, T A / A. These axioms were considered by Harvey Freidman and Michael Sheard in An axiomatic approach to self-referential truth [2]. They classified the above axioms and rules into nine maximally consistent sets. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 5 / 15

11 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

12 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

13 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

14 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

15 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

16 Ordinal Analyses Lower bounds U-Inf: n T Aṅ T x Ax T-Del: T T A T A T-Elim: T A /A T-Intro: A/T A Definition Let S 1 be Base T + U-Inf + T-Elim, and S 2 be Base T + U-Inf + T-Del + T-Intro + T-Elim. Denote by I (α) the formula A T TI( α, Ȧ). Sheard proved (in [4]) that S 1 α. I (α) I (ɛ α ). Moreover he showed S 1 = ϕ2. We have shown S 2 α. I (α) I (ϕnα) for each n. Proof. (Setch). From α. I (α) I (ϕnα) we get Prog β I (ϕn β) (with U-Inf). Thus, α. TI β (α, I (ϕn β)) I (ϕn α). Now by T-Intro, axioms of Base T and T-Del we have α. T TI β (α, I (ϕn β)) I (ϕn α). Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 6 / 15

17 Ordinal Analyses Infinitary Theories U-Inf: n T Aṅ T x Ax T-Elim: T A /A T-Del: T T A T A T-Intro: A/T A It is more interesting to find an upper bound for S 2. For this we need to tae a detour into infinitary logic. Definition (Inductive Definition of S 2 ) Define S 2 α,n (Cut). If (Ax.2.). (Ax.3.). Γ by (Ax.1), ( ), ( i ), (ω), ( ) and (T-Intro). If (T-Imp). If (T-Del). If (T-U-Inf). If if α, δ < β. Γ, A, δ, n Γ, T (A), T (A), Γ, A and A < then β, n Γ, Γ, T (A) if A is not an L T -sentence, β, m A and n < m then Γ, T (A), δ, n Γ, T (A), Γ, T (A B)) then β, n Γ, T (B), Γ, T T (A) then β, n Γ, T (A), Γ, T Aṁ for all m, β, n Γ, T ( x Ax), Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 7 / 15

18 Ordinal Analyses Infinitary Theories U-Inf: n T Aṅ T x Ax T-Elim: T A /A T-Del: T T A T A T-Intro: A/T A It is more interesting to find an upper bound for S 2. For this we need to tae a detour into infinitary logic. Definition (Inductive Definition of S 2 ) Define S 2 α,n (Cut). If (Ax.2.). (Ax.3.). Γ by (Ax.1), ( ), ( i ), (ω), ( ) and (T-Intro). If (T-Imp). If (T-Del). If (T-U-Inf). If if α, δ < β. Γ, A, δ, n Γ, T (A), T (A), Γ, A and A < then β, n Γ, Γ, T (A) if A is not an L T -sentence, β, m A and n < m then Γ, T (A), δ, n Γ, T (A), Γ, T (A B)) then β, n Γ, T (B), Γ, T T (A) then β, n Γ, T (A), Γ, T Aṁ for all m, β, n Γ, T ( x Ax), Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 7 / 15

19 Ordinal Analyses Infinitary Theories U-Inf: n T Aṅ T x Ax T-Elim: T A /A T-Del: T T A T A T-Intro: A/T A It is more interesting to find an upper bound for S 2. For this we need to tae a detour into infinitary logic. Definition (Inductive Definition of S 2 ) Define S 2 α,n (Cut). If (Ax.2.). (Ax.3.). Γ by (Ax.1), ( ), ( i ), (ω), ( ) and (T-Intro). If (T-Imp). If (T-Del). If (T-U-Inf). If if α, δ < β. Γ, A, δ, n Γ, T (A), T (A), Γ, A and A < then β, n Γ, Γ, T (A) if A is not an L T -sentence, β, m A and n < m then Γ, T (A), δ, n Γ, T (A), Γ, T (A B)) then β, n Γ, T (B), Γ, T T (A) then β, n Γ, T (A), Γ, T Aṁ for all m, β, n Γ, T ( x Ax), Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 7 / 15

20 Ordinal Analyses Infinitary Theories U-Inf: n T Aṅ T x Ax T-Elim: T A /A T-Del: T T A T A T-Intro: A/T A It is more interesting to find an upper bound for S 2. For this we need to tae a detour into infinitary logic. Definition (Inductive Definition of S 2 ) Define S 2 α,n (Cut). If (Ax.2.). (Ax.3.). Γ by (Ax.1), ( ), ( i ), (ω), ( ) and (T-Intro). If (T-Imp). If (T-Del). If (T-U-Inf). If if α, δ < β. Γ, A, δ, n Γ, T (A), T (A), Γ, A and A < then β, n Γ, Γ, T (A) if A is not an L T -sentence, β, m A and n < m then Γ, T (A), δ, n Γ, T (A), Γ, T (A B)) then β, n Γ, T (B), Γ, T T (A) then β, n Γ, T (A), Γ, T Aṁ for all m, β, n Γ, T ( x Ax), Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 7 / 15

21 Ordinal Analyses Infinitary Theories U-Inf: n T Aṅ T x Ax T-Elim: T A /A T-Del: T T A T A T-Intro: A/T A It is more interesting to find an upper bound for S 2. For this we need to tae a detour into infinitary logic. Definition (Inductive Definition of S 2 ) Define S 2 α,n (Cut). If (Ax.2.). (Ax.3.). Γ by (Ax.1), ( ), ( i ), (ω), ( ) and (T-Intro). If (T-Imp). If (T-Del). If (T-U-Inf). If if α, δ < β. Γ, A, δ, n Γ, T (A), T (A), Γ, A and A < then β, n Γ, Γ, T (A) if A is not an L T -sentence, β, m A and n < m then Γ, T (A), δ, n Γ, T (A), Γ, T (A B)) then β, n Γ, T (B), Γ, T T (A) then β, n Γ, T (A), Γ, T Aṁ for all m, β, n Γ, T ( x Ax), Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 7 / 15

22 Ordinal Analyses Infinitary Theories Definition The ran of A, A, is defined as follows. A = if A is an arithmetical literal or T (s) for some term s. A B = A B = x A = x A = A + 1. Theorem Cut Elimination S 2 ω α, n +1 Γ implies S 2 Γ. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 8 / 15

23 Ordinal Analyses Upper bounds For each n > and α define M n, α = N, { B : S2 α,m B for some m < n and α < α} and define M, α = N,. Lemma For each n define f n (α) = ϕn(ϕ1α). Then for every n < ω we have 1 If 2 If 3 If Γ then M n, fn(α) = Γ. T A then Γ then ϕ1 fn(α), p Γ. A for some p < n. Corollary If α < ϕω then α,n T A implies β,n A for some β < ϕω. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 9 / 15

24 Ordinal Analyses Upper bounds For each n > and α define M n, α = N, { B : S2 α,m B for some m < n and α < α} and define M, α = N,. Lemma For each n define f n (α) = ϕn(ϕ1α). Then for every n < ω we have 1 If 2 If 3 If Γ then M n, fn(α) = Γ. T A then Γ then ϕ1 fn(α), p Γ. A for some p < n. Corollary If α < ϕω then α,n T A implies β,n A for some β < ϕω. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 9 / 15

25 Ordinal Analyses Upper bounds For each n > and α define M n, α = N, { B : S2 α,m B for some m < n and α < α} and define M, α = N,. Lemma For each n define f n (α) = ϕn(ϕ1α). Then for every n < ω we have 1 If 2 If 3 If Γ then M n, fn(α) = Γ. T A then Γ then ϕ1 fn(α), p Γ. A for some p < n. Corollary If α < ϕω then α,n T A implies β,n A for some β < ϕω. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 9 / 15

26 Ordinal Analyses Upper bounds Thus, we have Lemma If S 2 A then S 2 and α,n A for some α < ϕω. Theorem Let A be an arithmetical sentence, then S 2 A implies PA + TI(<ϕω) A. Hence Corollary (T-Elimination for S 2 ) S 2 = ϕω. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 1 / 15

27 Ordinal Analyses Upper bounds Thus, we have Lemma If S 2 A then S 2 and α,n A for some α < ϕω. Theorem Let A be an arithmetical sentence, then S 2 A implies PA + TI(<ϕω) A. Hence Corollary (T-Elimination for S 2 ) S 2 = ϕω. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 1 / 15

28 The theory E U-Inf: n T Aṅ T x Ax, T-Elim: T A /A, T-Cons: T-Del: T T A T A, T-Intro: A/T A (T A T A ) The bounds for S 2 were fairly easy to establish. However, this is not the case for all nine of the theories we considered. For example, E is given by Base T + T-Del + U-Inf + T-Cons + T-Intro + T-Elim. The upper bound E is not so clear because we no longer have Cut-Elimination in the corresponding infinitary system. However, we can embed E in a small extension of ID 1. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

29 The theory E U-Inf: n T Aṅ T x Ax, T-Elim: T A /A, T-Cons: T-Del: T T A T A, T-Intro: A/T A (T A T A ) The bounds for S 2 were fairly easy to establish. However, this is not the case for all nine of the theories we considered. For example, E is given by Base T + T-Del + U-Inf + T-Cons + T-Intro + T-Elim. The upper bound E is not so clear because we no longer have Cut-Elimination in the corresponding infinitary system. However, we can embed E in a small extension of ID 1. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

30 The theory E ID + 1 ID 1 is the theory extending PRA in which for each arithmetical formula A L + P with only one free variable the language is augmented by an additional predicate symbol I A and we have the axioms u. A(u, I A ) I A (u), (Ax.I A.1) u[a(u, F ) F (u)] u[i A (u) F (u)], (Ax.I A.2) for each formula F containing only positive occurrences of predicates I B for B L + P and induction for formulae where fixed-point predicates occur positively. We define ID + 1 to be ID 1 with, as an additional axiom, u[a(u, F ) F (u)] u[i A (u) F (u)], (Ax.I A.3) if A L + P is Σ 2 and F is any formula which is Σ 1 or Π 1 in I A and I A Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

31 The theory E ID + 1 ID 1 is the theory extending PRA in which for each arithmetical formula A L + P with only one free variable the language is augmented by an additional predicate symbol I A and we have the axioms u. A(u, I A ) I A (u), (Ax.I A.1) u[a(u, F ) F (u)] u[i A (u) F (u)], (Ax.I A.2) for each formula F containing only positive occurrences of predicates I B for B L + P and induction for formulae where fixed-point predicates occur positively. We define ID + 1 to be ID 1 with, as an additional axiom, u[a(u, F ) F (u)] u[i A (u) F (u)], (Ax.I A.3) if A L + P is Σ 2 and F is any formula which is Σ 1 or Π 1 in I A and I A Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

32 The theory E E and inductive definitions Theorem There are formulae A n (u, P + ) such that E C implies there is an n such that ID + 1 I n ( C ). Theorem Every arithmetical consequence of E is a theorem of ID + 1. Proof. Let A be a model for the first-order part of ID + 1. Using A we may then build a hierarchy of L T -structures M = A, ; M n+1 = A, I n. with the property that M n = I n. Now, if E A then M n = A for some n. Hence A = A. But A was arbitrary. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

33 The theory E E and inductive definitions Theorem There are formulae A n (u, P + ) such that E C implies there is an n such that ID + 1 I n ( C ). Theorem Every arithmetical consequence of E is a theorem of ID + 1. Proof. Let A be a model for the first-order part of ID + 1. Using A we may then build a hierarchy of L T -structures M = A, ; M n+1 = A, I n. with the property that M n = I n. Now, if E A then M n = A for some n. Hence A = A. But A was arbitrary. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

34 The theory E E and inductive definitions Theorem There are formulae A n (u, P + ) such that E C implies there is an n such that ID + 1 I n ( C ). Theorem Every arithmetical consequence of E is a theorem of ID + 1. Proof. Let A be a model for the first-order part of ID + 1. Using A we may then build a hierarchy of L T -structures M = A, ; M n+1 = A, I n. with the property that M n = I n. Now, if E A then M n = A for some n. Hence A = A. But A was arbitrary. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

35 The theory E E and inductive definitions And so, Theorem ϕω = ID 1 E ID + 1. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15

36 References References Than you. A. Cantini. A theory of formal truth arithmetically equivalent to ID1. Journal of Symbolic Logic, 55(1): , March 199. H. Friedman and M. Sheard. An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic, 33:1 21, V. Halbach. A system of complete and consistent truth. Notre Dame Journal of Formal Logic, 35(3), M. Sheard. Wea and strong theories of truth. Studia Logica, 68:89 11, 21. Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July / 15