ar X iv :1 80 9. 10 53 9v 13 [ m at h. L O ] 3 F eb 2 02 0 Theories of truth for countable languages which conform to classical logic Seppo Heikkilä ⋆ Department of Mathematical Sciences, University of Oulu BOX 3000, FIN-90014, Oulu, Finland Abstract Every countable language which conforms to classical logic is shown to have an extension which conforms to classical logic, and has a definitional theory of truth. That extension has a semantical theory of truth, if every sentence of the object language is valuated by its meaning either as true or as false. These theories contain both a truth predicate and a non-truth predicate. Theories are equivalent when the sentences of the object language are valuated by their meanings. 1 Introduction Based on 'Chomsky Definition' (cf. [2]) we assume that a language is a nonempty countable set of sentences with finite length, and formed by a countable set of elements. A theory of syntax is also assumed to provide a language with rules to construct well-formed sentences, formulas etc. A language is said to conform to classical logic if it has, or if it can be extended to have at least the following properties ('iff' means 'if and only if'): (i) It contains logical symbols ¬ (not), ∨ (or), ∧ (and), → (if...then), ↔ (iff), ∀ (for all) and ∃ (exists), and the following sentences: If A and B are (denote) sentences of the language, so are ¬A, A ∨ B, A∧B, A → B and A ↔ B. If P (x1, . . . , xm), m ≥ 1, is a formula of the language with m free variables, then P is called a predicate with arity m and domain DP = D 1 P × * * *×D m P , where each D i P is a subset of a set D of objects, called the domain of discourse, if the following properties hold. (p1) Every object of D is named by a term. Denote by N iP the set of those terms which name the objects of DiP , i = 1, . . . , m. (p2) P (b1, . . . , bm) is a sentence of that language obtained from P (x1, . . . , xm) by substituting for each i = 1, . . . , m a term bi of N i P for xi in every free occurrence of xi in P . If P is a predicate with arity m ≥ 1, then the sentences q1x1 . . . qmxmP (x1, . . . , xm), where each qi is either ∀ or ∃, are in the language. ¬P is also a predicate with arity m and domain DP . (ii) The sentences of that language are so valuated as true or as false that the following rules of classical logic are valid: If A and B denote sentences of the language, then A is true iff ¬A is false, and A is false iff ¬A is true; A ∨ B is true iff A or B is true, and false iff A and B are false; A ∧ B is true iff A and B are true, and false iff A or B is false; A → B is true iff A is false or B is true, and false iff A is Mathematics Subject Classification: 03B10, 03B65, 03D80, 91F20, 97M80 Keywords: language, countable, sentence, valuation, true, false, meaning, truth predicate, logic, classical 1 true and B is false; A ↔ B is true iff A and B are both true or both false, and false iff A is true and B is false or A is false and B is true. If P is a predicate with arity m, then the sentence of the form q1x1 . . . qmxmP (x1, . . . , xm) is true iff the sentence P (b1, . . . , bm) is true for all bi ∈ N i P when qi is ∀, and for some bi ∈ N i P when qi is ∃. q1x1 . . . qmxmP (x1, . . . , xm) is false iff the sentence P (b1, . . . , bm) is false for all bi ∈ N i P when qi is ∃, and for some bi ∈ N i P when qi is ∀. (iii) The language is bivalent, i.e., every sentence of it is either true or false. Every countable and bivalent first-order language with or without identity conforms to classical logic. A classical example is the language of arithmetic in its standard interpretation. We say that a language has a theory of truth if truth values are assigned to its sentences, and if it contains a predicate T which satisfies T -rule: T (⌈A⌉) ↔ A is true for every sentence A of the language. Term ⌈A⌉ which names the sentence A is defined below. A predicate T which satisfies T -rule is called a truth predicate. A theory of truth is said to be definitional if truth values are defined for sentences, and semantical if truth values of sentences are determined by their meanings. Main results of this paper are: Every countable language which conforms to classical logic has an extension which has properties (i)– (iii), and has a definitional theory of truth. That extension has a semantical theory of truth if every sentence of the object language is valuated by its meaning either as true or as false. These theories of truth contain truth and non-truth predicates. Theories are equivalent when the sentences of the object language are valuated by their meanings. 2 Extended languages Assume that an object language L0 conforms to classical logic, and is without a truth predicate. L0 has by definition an extension which has properties (i) – (iii). That extension, denoted by L, is called a basic extension of L0. The language LT is formed by adding to L extra formulas T (x) and ¬T (x), and sentences T (n) and ¬T (n), where n goes through all numerals which denote numbers n ∈ N0 = {0, 1, 2, . . .} (0=0, 1=S0, 2=SS0,. . . ). Neither valuation nor meaning is yet attached to these sentences. Numerals are added, if necessary, to terms of LT . Choose a Gödel numbering to sentences of LT (see Wikipedia). The Gödel number of a sentence denoted by A is denoted by #A, and the numeral of #A by ⌈A⌉, which names the sentence A. If P is a predicate of L with arity m, then P (b1, . . . bm) is a sentence of L for each (b1, . . . bm) ∈ NP = NP1 × * * * × N m P , and ⌈P (b1, . . . bm)⌉ is the numeral of its Gödel number. Thus T (⌈P (b1, . . . , bm)⌉) and ¬T (⌈P (b1, . . . , bm)⌉) are sentences of LT for each (b1, . . . , bm) ∈ NP , so that they are determined by predicates of LT having the domain Dp of P . Denote these predicates by T (⌈P (ẋ1, . . . , ẋm)⌉) and ¬T (⌈P (ẋ1, . . . , ẋm)⌉), and add them to LT . Notation P (ẋ1, . . . , ẋm) stands for the result of formally replacing variables x1, . . . xm of P (x1, . . . , xm) by terms of NP (cf. [4]). Add to the language LT sentences ∀xT1(x), ∃xT1(x), ∀xT1(⌈T2(ẋ)⌉) and ∃xT1(⌈T2(ẋ)⌉), where T1, T2 ∈ {T,¬T}, and sentences q1x1 . . . qmxmT (⌈P (ẋ1, . . . , ẋm)⌉) and q1x1 . . . qmxm¬T (⌈P (ẋ1, . . . , ẋm)⌉) for each predicate P of L with arity m ≥ 1 and for each m-tuple q1, . . . , qm, where qi's are ∀ or ∃. The so obtained extension of the language LT is denoted by L0. 2 When a language Ln, n ∈ N0, is defined, let Ln+1 be a language which is formed by adding to Ln those of the following sentences which are not in Ln: ¬A, A ∨ B, A ∧ B, A → B and A ↔ B, where A and B are sentences of Ln. The language L is defined as the union of languages Ln, n ∈ N0. Extend the Gödel numbering of the sentences of LT to those of L, and denote by D the set of Gödel numbers of the sentences of L. Our main goal is to extract from L a sublanguage which under suitable valuations of its sentences has properties (i)–(iii) given in Introduction, and has a theory of truth. At first we define some subsets of L. Denote by Pm the set of those predicates of L which have arity m, P = ⋃ ∞ m=1P m, and { Zm3 = {q1x1 . . . qmxmT (⌈P (ẋ1, . . . , ẋm)⌉) : P ∈ P m and q1x1 . . . qmxmP (x1, . . . , xm) is true}; Zm4 = {q1x1 . . . qmxm¬T (⌈P (ẋ1, . . . , ẋm)⌉) : P ∈ P m and q1x1 . . . qmxm¬P (x1, . . . , xm) is true}. (2.1) Define subsets Z1(U), Z2(U), U ⊂ D, and Zi, i = 1 . . . 4, of L by              Z1(U) = {T (n): n = ⌈A⌉, where A is a sentence of L and #A is in U}, Z2(U) = {¬T (n): n = ⌈A⌉, where A is a sentence of L and #[¬A] is in U}, Z1 = {¬∀xT (x), ∃xT (x),¬∀x¬T (x), ∃x¬T (x)}, Z2 = {¬(∀xT1(⌈T2(ẋ)⌉)), ∃xT1(⌈T2(ẋ)⌉), T1, T2 ∈ {T,¬T}, Z3 = ⋃ ∞ m=1 Z m 3 , Z4 = ⋃ ∞ m=1 Z m 4 . (2.2) Subsets Ln(U), n ∈ N0, of L are defined recursively as follows. L0(U) = { Z = {A : A is a true sentence of L} if U = ∅ (the empty set), Z ∪ Z1(U) ∪ Z2(U) ∪ Z1 ∪ Z2 ∪ Z3 ∪ Z4 if ∅ ⊂ U ⊂ D. (2.3) When a subset Ln(U) of L is defined for some n ∈ N0, and when A and B are sentences of L, denote                                L0n(U) = {¬(¬A) : A is in Ln(U)}, L1n(U) = {A ∨ B : A and B, or A and ¬B, or ¬A and B are in Ln(U)}, L2n(U) = {A ∧ B : A and B are in Ln(U)}, L3n(U) = {A → B : ¬A and B, or ¬A and ¬B, or A and B are in Ln(U)}, L4n(U) = {A ↔ B : A and B, or ¬A and ¬B are in Ln(U)}, L5n(U) = {¬(A ∨B) : ¬A and ¬B are in Ln(U)}, L6n(U) = {¬(A ∧B) : ¬A and B, or ¬A and ¬B, or A and ¬B are in Ln(U)}, L7n(U) = {¬(A → B) : A and ¬B are in Ln(U)}, L8n(U) = {¬(A ↔ B) : A and ¬B, or ¬A and B are in Ln(U)}, (2.4) and define Ln+1(U) = Ln(U) ∪ 8 ⋃ k=0 Lkn(U). (2.5) 3 The above constructions imply that Lkn(U) ⊆ L k n+1(U) and Ln(U) ⊂ Ln+1(U) ⊂ L for all n ∈ N0 and k = 0, . . . , 8. Define a subset L(U) of L by L(U) = ∞ ⋃ n=0 Ln(U). (2.6) 3 Properties of consistent subsets of D Recall that D denotes the set of Gödel numbers of the sentences of L. When U is a subset of D, denote by G(U) the set of Gödel numbers of the sentences of L(U) defined by (2.6): G(U) = {#A : A is a sentence of L(U)}. (3.1) A subset U of D is called consistent if both #A and #[¬A] are not in U for any sentence A of L . Lemma 3.1. Let U be a consistent subset of D. Then for no sentence A of L both A and ¬A belong to L(U), and G(U) is consistent. Proof. At first we show that there is no sentence A in L such that both A and ¬A belong to L0(U). If U = ∅, then L0(U) is by (2.3) the set Z of true sentences of L. If A is a sentence of L, then only one of the sentences A and ¬A is true, and hence in Z = L0(U), since L has properties (i)–(iii). Assume next that U is nonempty. As a consistent set U is a proper subset of D. Let n be a numeral. If T (n) is in L0(U), it is in Z1(U), so that, by (2.2), n = ⌈A⌉, where #A is in U . Since U is consistent, then #[¬A] is not in U . Thus, by (2.2), ¬T (n) is not in Z2(U), and hence not in L0(U). This result implies also that T (n) is not in L0(U) if ¬T (n) is in L0(U). (2.2) and (2.3) imply that sentences ∃xT (x), ¬∀xT (x), ¬∀x¬T (x) and ∃x¬T (x) are in Z1, and hence in L0(U), but their negations are not in L0(U). By the definitions (2.2) and (2.3) of Z2 neither both ∃xT1(⌈T2(ẋ)⌉) and ¬(∃xT1(⌈T2(ẋ)⌉)), nor both ∀xT1(⌈T2(ẋ)⌉) and ¬(∀xT1(⌈T2(ẋ)⌉)), are in L0(U) for any T1, T2 ∈ {T,¬T}. By the definitions (2.1), (2.2) and (2.3) the sentences of Z3 and Z4, but not their negations, are in L0(U). The above proof shows that for no sentence A of L both A and ¬A belong to L0(U). Make the induction hypothesis: (h0) For no sentence A of L both A and ¬A belong to Ln(U). Applying (h0) and (2.4) we obtain the following results. (h0) and the definition of L0n(U) imply that if a sentence A is in Ln(U), then none of the odd-tuple negations of A are in Ln+1(U), and if ¬A is in Ln(U), then none of the even-tuple negations of A are in Ln+1(U). If A ∨ B is in Ln+1(U), it is in L 1 n(U), whence A or B is in Ln(U). If ¬(A ∨ B) is in Ln+1(U), it is in L5n(U), in which case ¬A and ¬B are in Ln(U). Thus A ∨ B and ¬(A ∨ B) are not both in Ln+1(U), for otherwise both A and ¬A or both B and ¬B are in Ln(U), contradicting with (h0). A ∧ B and ¬(A ∧ B) cannot both be in Ln+1(U), for otherwise A ∧ B is in L 2 n(U), i.e., both A and B are in Ln(U), and ¬(A∧B) is in L 6 n(U), i.e., at least one of ¬A and ¬B is in Ln(U). Thus both A and ¬A or both B and ¬B are in Ln(U), contradicting with (h0). If A → B is in Ln+1(U), it is in L 3 n(U), so that ¬A or B is in Ln(U). If ¬(A → B) is in Ln+1(U), it is in L7n(U), whence both A and ¬B are in Ln(U). Because of these results and (h0) the sentences A → B and ¬(A → B) are not both in Ln+1(U). 4 If A ↔ B is Ln+1(U), it is in L 4 n(U), in which case both A and B or both ¬A and ¬B are in Ln(U). If ¬(A ↔ B) is in Ln+1(U), it is in L 8 n(U), whence both A and ¬B or both ¬A and B are in Ln(U). Thus both A ↔ B and ¬(A ↔ B) cannot be in Ln+1(U), for otherwise both A and ¬A or both B and ¬B are in Ln(U), contradicting with (h0). The above results and the induction hypothesis (h0) imply that for no sentence A of L both A and ¬A belong to Ln+1(U) = Ln(U) ∪ ⋃8 k=0 L k n(U). Since (h0) is proved when n = 0, it is by induction valid for every n ∈ N0. If A and ¬A are in L(U), then A is by (2.6) in Ln1(U) for some n1 ∈ N0, and ¬A is in Ln2(U) for some n2 ∈ N0. Then both A and ¬A are in Ln(U) when n = max{n1, n2}. This is impossible, because (h0) is proved for every n ∈ N0. Thus A and ¬A cannot both be in L(U) for any sentence A of L. The above result and (3.1) imply that there is no sentence A in L such that both #A and #[¬A] are in G(U). Thus G(U) is consistent. Lemma 3.2. Assume that U and V are consistent subsets of D, and that V ⊆ U . Then L(V ) ⊆ L(U) and G(V ) ⊆ G(U). Proof. As consistent sets V and U are proper subsets of D. At first we show that L0(V ) ⊆ L0(U). If V = ∅, then L0(V ) = Z ⊆ L0(U) by (2.3). Assume next that V is nonempty. Thus also U is nonempty. Let A be a sentence of L. Definition (2.3) of L0(U) implies that A is in L0(U) and also in L0(V ) iff A is in Z. Let n be a numeral. If T (n) is in L0(V ), it is in Z1(V ), so that n = ⌈A⌉, where #A is in V . Because V ⊆ U , then #A is also in U , whence T (n) is in Z1(U), and hence in L0(U). If ¬T (n) is in L0(V ), it is in Z2(V ), in which case n = ⌈A⌉, where #[¬A] is in V . Since V ⊆ U , then #[¬A] is also in U , whence ¬T (n) is in Z2(U), and hence in L0(U). Because U and V are nonempty and proper subsets of D, then Z1, Z2, Z3 and Z4 are in L0(U) and in L0(V ) by (2.3). The above results imply that L0(V ) ⊆ L0(U). Make the induction hypothesis: (h1) Ln(V ) ⊆ Ln(U) for some n ∈ N0. It follows from (2.4) and (h1) that L k n(V ) ⊆ L k n(U) for each k = 0, . . . , 8. Thus Ln+1(V ) = Ln(V ) ∪ 8 ⋃ k=0 Lkn(V ) ⊆ Ln(U) ∪ 8 ⋃ k=0 Lkn(U) = Ln+1(U). (h1) is proved when n = 0, whence it is by induction valid for every n ∈ N0. If A is in L(V ), it is by (2.6) in Ln(V ) for some n ∈ N0. Thus A is in Ln(U) by (h1), and hence in L(U). Consequently, L(V ) ⊆ L(U). If #A is in G(V ) then A is in L(V ) by (3.1). Thus A is in L(U), so that #A is in G(U) by (3.1). This shows that G(V ) ⊆ G(U). Denote by C the family of consistent subsets of D. In the formulation and the proof of Theorem 3.1 transfinite sequences indexed by ordinals are used. A transfinite sequence (Uλ)λ<α of C is said to be increasing if Uμ ⊆ Uν whenever μ < ν < α, and strictly increasing if Uμ ⊂ Uν whenever μ < ν < α. Lemma 3.3. Assume that (Uλ)λ<α is a strictly increasing sequence of C. Then (a) (G(Uλ))λ<α is an increasing sequence of C. (b) The union ⋃ λ<α G(Uλ) is consistent. 5 Proof. Since Uμ ⊂ Uν when μ < ν < α, it follows from Lemma 3.2 that G(Uμ) ⊆ G(Uν) when μ < ν < α, whence the sequence (G(Uλ))λ<α is increasing. Consistency of the sets G(Uλ), λ < α, follows from Lemma 3.1 because the sets Uλ, λ < α, are consistent. This proves (a). To prove that the union ⋃ λ<α G(Uλ) is consistent, assume on the contrary that there exists such a sentence A in L that both #A and #[¬A] are in ⋃ λ<α G(Uλ). Thus there exist μ, ν < α such that #A is in G(Uμ) and #[¬A] is in G(Uν). Because G(Uμ) ⊆ G(Uν) or G(Uν) ⊆ G(Uμ), then both #A and #[¬A] are in G(Uμ) or in G(Uν). But this is impossible, since both G(Uμ) and G(Uν) are consistent. Thus, the set ⋃ λ<α G(Uλ) is consistent. Now we are ready to prove the following Theorem. Theorem 3.1. Let W denote the set of Gödel numbers of true sentences of L. We say that a transfinite sequence (Uλ)λ<α of C is a G-sequence if it has the following properties. (G) (Uλ)λ<α is strictly increasing, U0 = W , and if 0 < μ < α, then Uμ = ⋃ λ<μ G(Uλ). Then the longest G-sequence exists, and it has the last member. This member is the smallest consistent subset U of D satisfying U = G(U). Proof. W is consistent, since L has properties (i)-(iii). At first we show that G-sequences are nested: (1) Assume that (Uλ)λ<α and (Vλ)λ<β are G-sequences. Then Uλ = Vλ when λ < min{α, β}. U0 = W = V0 by (G). Make the induction hypothesis: (h) There exists an ordinal ν which satisfies 0 < ν < min{α, β} such that Uλ = Vλ for each λ < ν. It follows from (h) and (G) that Uν = ⋃ λ<ν G(Uλ) = ⋃ λ<ν G(Vλ) = Vν . Since U0 = V0, then (h) holds when ν = 1. These results imply (1) by transfinite induction. Let (Uλ)λ<α be a G-sequence. Defining f(0) = minU0, f(λ) = min(Uλ \ Uλ−1), 0 < λ < α, and f(α) = min(D \ ⋃ λ<α Uλ), we obtain a bijection f from [0, α] to a subset of N0. Thus α is a countable ordinal. Consequently, the set Γ of those ordinals α for which (Uλ)λ<α is a G-sequence is bounded from above by the smallest uncountable ordinal. Denote by γ the least upper bound of Γ. To show that γ is a successor, assume on the contrary that γ is a limit ordinal. Given any μ < γ, then ν = μ + 1 and α = ν + 1 are < γ. (Uλ)λ<α is a G-sequence, whence Uμ = ⋃ λ<μ G(Uλ), and Uμ ⊂ Uμ+1. Thus (Uλ)λ<γ has properties (G) when α = γ, so that (Uλ)λ<γ is a G-sequence. Denote Uγ = ⋃ λ<γ G(Uλ). Uγ is consistent by Lemma 3.3(b). Because Uμ ⊂ Uν = ⋃ λ<ν G(Uλ) ⊆ Uγ for each μ < γ, then (Uλ)λ<γ+1 is a G-sequence. But this contradicts with the choice of γ. Thus γ is a successor, say γ = α + 1. If λ < α, then Uλ ⊂ Uα, so that G(Uλ) ⊆ G(Uα). Then Uα = ⋃ λ<α G(Uλ) ⊆ ⋃ λ<γ G(Uλ) = G(Uα), whence Uα ⊆ G(Uα). Moreover, Uα = G(Uα), for otherwise Uα ⊂ G(Uα) = ⋃ λ<γ G(Uλ) = Uγ , and (Uλ)λ<γ+1 would be a G-sequence. Consequently, (Uλ)λ<γ is the longest G-sequence, Uα is its last member, and Uα = G(Uα). Let U be a consistent subset of D satisfying U = G(U). Then U0 = W = G(∅) ⊆ G(U) = U . Make the induction hypothesis: (h2) There exists an ordinal μ which satisfies 0 < μ < γ such that Uλ ⊆ U for each λ < μ. 6 Then G(Uλ) ⊆ G(U) for each λ < μ, whence Uμ = ⋃ λ<μ G(Uλ) ⊆ G(U) = U . Thus, by transfinite induction, Uμ ⊆ U for each μ < γ. In particular, Uα ⊆ U . This proves the last assertion of Theorem. 4 Language LT and its properties Let L0 be a language which conforms to classical logic and has not a truth predicate. Let L, P and D be as in Section 2, and let L(U) and G(U), U ⊂ D, be defined by (2.6) and (3.1). Define F (U) = {A : ¬A ∈ L(U)}. (4.1) Recall that a subset U of D is consistent if there is no sentence A in L such that both #A and #[¬A] are in U . By Theorem 3.1 the smallest consistent subset of D which satisfies U = G(U) exists. Definition 4.1. Let U be the smallest consistent subset of D which satisfies U = G(U). Denote by LT the language formed by the object language L0, the predicates of P, the sentences of L(U) and F (U), formulas T (x) and ¬T (x), corresponding predicates T and ¬T with their domain DT and the set NT of terms defined by DT = L(U) ∪ F (U) and NT = {n : n = ⌈A⌉ , where A is in DT}, (4.2) and predicates T1(⌈T2(ẋ)⌉), and T1(⌈P (ẋ1, . . . , ẋm)⌉), where T1, T2 ∈ ∪{T,¬T} and P ∈ P with arity m ≥ 1. A valuation is defined for sentences of LT as follows. (I) A sentence of LT is valuated as true iff it is in L(U), and as false iff it is in F (U). Lemma 4.1. The language LT defined by Definition 4.1 and valuated by (I) is bivalent. Proof. The subsets L(U) and F (U) of the sentences of LT are disjoint. For otherwise there is a sentence A of LT which is in L(U) ∩ F (U). Then A is in L(U), and by the definition (4.1) of F (U) also ¬A is in L(U). But this is impossible by Lemma 3.1. Consequently, L(U) ∩ F (U) = ∅. If A is a sentence of LT , then it is in L(U) or in F (U). If A is true, it is in L(U), but not in F (U), and hence not false, because L(U) ∩ F (U) = ∅. Similarly, if A is false, it is in F (U), but not in L(U), and hence not true. Consequently, A is either true or false, so that LT is bivalent. Lemma 4.2. Let LT be defined by Definition 4.1 and valuated by (I). Then a sentence of the basic extension L of L0 is true (respectively false) in the valuation (I) iff it is true (respectively false) in the valuation of L. Proof. Let A denote a sentence of L. A is true in the valuation (I) iff A is in L(U) iff (by the construction of L(U)) A is in Z iff A is true in the valuation of L. A is false in the valuation (I) iff A is in F (U) iff (by (4.1)) ¬A is in L(U) iff (¬A is a sentence of L) ¬A is in Z iff ¬A is true in the valuation of L iff (L has properties (i)–(iii)) A is false in the valuation of L. Lemma 4.3. The language LT defined by Definition 4.1 and valuated by (I) has properties (i) and (ii) given in Introduction. 7 Proof. Unless otherwise stated, 'true' means true in the valuation (I), and 'false' means false in the valuation (I). The construction of L(U) and the definition (4.1) of F (U) imply that LT has properties (i). As for properties (ii) we at first derive the following auxiliary rule. (t0) Double negation: If A is a sentence of LT , then ¬(¬A) is true iff A is true. To prove (t0), assume first that ¬(¬A) is true. Then it is in L(U), and hence, by (2.6), in Ln(U) for some n ∈ N0. If ¬(¬A) is in L0(U) then it by (2.3) in Z. Thus ¬(¬A) is true in the valuation of L. Then (negation rule is valid in L) ¬A is false in the valuation of L, which implies that A is true in the valuation of L. Thus A is by (2.3) in Z ⊂ L0(U) ⊂ L(U), whence A is true. Assume next that n ∈ N0 is the smallest number for which ¬(¬A) is in Ln+1(U). It then follows from (2.4) and (2.5) that ¬(¬A) is in L0n(U), so that A is in Ln(U), and hence in L(U), i.e., A is true. Thus A is true if ¬(¬A) is true. Conversely, assume that A is true. Then A is in L(U), so that A is in Ln(U) for some n ∈ N0. Thus ¬(¬A) is in L0n(U), and hence in Ln+1(U). Consequently, ¬(¬A) is in L(U), whence ¬(¬A) is true. This concludes the proof of (t0). Rule (t0) is applied to prove (t1) Negation: A is true iff ¬A is false, and A is false iff ¬A is true. Let A be a sentence of LT . Then A is true iff (by (t0)) ¬(¬A) is true iff ¬(¬A) is in L(U) iff (by (4.1)) ¬A is in F (U) iff ¬A is false. A is false iff A is in F (U) iff (by (4.1)) ¬A is in L(U) iff ¬A is true. Thus (t1) is satisfied. Next we prove the following rule. (t2) Conjunction: A ∧B is true iff A and B are true. A ∧ B is false iff A or B is false. Let A and B be sentences of LT . If A and B are true, i.e., A and B are in L(U), there is by (2.6) an n ∈ N0 such that A and B are in Ln(U). Thus A ∧B is by (2.4) in L 2 n(U), and hence in L(U), so that A ∧ B is true. Conversely, assume that A ∧ B is true, or equivalently, A ∧ B is in L(U). Then there is by (2.6) an n ∈ N0 such that A∧B is in Ln(U). If A∧B is in L0(U), it is in Z. Thus A∧B is true in the valuation of L. Because L has property (ii), then A and B are true in the valuation of L, and hence also in the valuation (I) by Lemma 4.2. Assume next that n ∈ N0 is the smallest number for which A∧B is in Ln+1(U). Then A∧B is by (2.4) in L2n(U), so that A and B are in Ln(U), and hence in L(U), i.e., A and B are true. The above reasoning proves that A ∧ B is true iff A and B are true. This result and the bivalence of LT , proved in Lemma 4.1, imply that A∧B is false iff A or B is false. Consequently, rule (t2) is valid. The proofs of the following rules are similar to the above proof of (t2). (t3) Disjunction: A ∨B is true iff A or B is true. A ∨ B false iff A and B are false. (t4) Conditional: A → B is true iff A is false or B is true. A → B is false iff A is true and B is false. (t5) Biconditional: A ↔ B is true iff A and B are both true or both false. A ↔ B is false iff A is true and B is false or A is false and B is true. Next we show that if T1 ∈ {T,¬T} then ∃xT1(x) and ∀xT1(x) have the following properties. 8 (t6) ∃xT1(x) is true iff T1(n) is true for some n ∈ NT , and false iff T1(n) is false for every n ∈ NT . (t7) ∀xT1(x) is true iff T1(n) is true for every n ∈ NT , and false iff T1(n) is false for some n ∈ NT . To simplify proofs we derive results which imply that T is a truth predicate and ¬T is a non-truth predicate for LT . Let A denote a sentence of LT . The valuation (I), rule (t1), the definitions of Z1(U), Z2(U) and G(U), and the assumption U = G(U) imply that A is true iff A is in L(U) iff #A is in G(U) = U iff T (⌈A⌉) is in Z1(U) ⊂ L(U) iff T (⌈A⌉) is true iff ¬T (⌈A⌉) is false. A is false iff A is in F (U) iff ¬A is in L(U) iff #[¬A] is in G(U) = U iff ¬T (⌈A⌉) is in Z2(U) ⊂ L(U) iff ¬T (⌈A⌉) is true iff T (⌈A⌉) is false. The above results imply that the following results are valid for every sentence A ∈ LT . (T) A is true iff T (⌈A⌉) is true iff ¬T (⌈A⌉) is false. A is false iff T (⌈A⌉) is false iff ¬T (⌈A⌉) is true. Consider the validity of (t6) and (t7) when T1 is T . Because U is nonempty, then ∃xT (x) is in L0(U) by (2.2) and (2.3), and hence in L(U) by (2.6). Thus ∃xT (x) is by (I) a true sentence of LT . T (⌈A⌉) is true iff (by (T)) A is true iff (by (I)) A is in L(U). Thus T (n) is true for some n ∈ NT . The above results imply that ∃xT (x) is true iff T (n) is true for some n ∈ NT . In view of this result and the bivalence of LT , one can infer that ∃xT (x) is false iff T (n) is false for every n ∈ NT . This concludes the proof of (p6) when T1 is T . ¬∀xT (x) is in Z1 ⊂ L0(U), and hence in L(U), so that it is true. Thus ∀xT (x) is false by (t1). T (⌈A⌉) is false iff (by (T)) A is false iff (by (I)) A is in F (U). Thus T (n) is false for some n ∈ NT . Consequently, ∀xT (x) is false iff T (n) is false for some n ∈ NT . This result and the bivalence of LT imply that ∀xT (x) is true iff T (n) is true for every n ∈ NT . This proves (t7) when T1 is T . To show that (t6) is valid when T1 is ¬T , notice first that ∃x¬T (x) is in Z1 ⊂ L0(U), and hence in L(U), whence it is true. ¬T (⌈A⌉) is true iff (by (t1)) T (⌈A⌉) is false iff (by (T)) A is false iff (by (I)) A is in F (U). Thus ¬T (n) is true for some n ∈ NT . Consequently, ∃x¬T (x) is true iff ¬T (n) is true for some n ∈ NT . This result and the bivalence of LT imply that ∃x¬T (x) is false iff ¬T (n) is false for every n ∈ NT . This concludes the proof of (t6) when T1 is ¬T . Next we prove (t7) when T1 is ¬T . ¬∀x¬T (x) is in Z1 ⊂ L0(U), and hence in L(U), so that it is true. Thus ∀x¬T (x) is false by (t1). ¬T (⌈A⌉) is false iff (by (t1)) T (⌈A⌉) is true iff (by (T)) A is true iff (by (I)) A is in L(U). Thus ¬T (n) is false for some n ∈ NT . From these results it follows that ∀x¬T (x) is false iff ¬T (n) is false for some n ∈ NT . This result and bivalence of LT imply that ∀x¬T (x) is true iff ¬T (n) is true for all n ∈ NT . Thus (t7) is valid when T1 is ¬T . Next we show that the following rules are valid when T1, T2 ∈ {T,¬T}. (tt6) ∃xT1(⌈T2(ẋ)⌉) is true iff T1(⌈T2(n)⌉) is true for some n ∈ NT . ∃xT1(⌈T2(ẋ)⌉) is false iff T1(⌈T2(n)⌉) is false for every n ∈ NT ; (tt7) ∀xT1(⌈T2(ẋ)⌉) is true iff T1(⌈T2(n)⌉) is true for every n ∈ NT . ∀xT1(⌈T2(ẋ)⌉) is false iff T1(⌈T2(n)⌉) is false for some n ∈ NT . 9 The sentences ∃xT1(⌈T2(ẋ)⌉), where T1, T2 ∈ {T,¬T} are in Z2, whence they are in L(U) and hence true. Applying (T) one can show that in every case there exists an n ∈ NT so that T1(⌈T2(n)⌉) is true (n = ⌈A⌉, where A, depending on the case, is in L(U) or in F (U)). These results imply truth part of (tt6) when T1 and T2 are in {T,¬T}. Falsity part in (tt6) is then valid by bivalence of LT . ∀xT1(⌈T2(ẋ)⌉) is false because its negation is in Z2 and hence true. Using (T) it is easy to show that T1(⌈T2(n)⌉) is false for some n ∈ NT whenever T1, T2 ∈ {T,¬T}. This proves the falsity part of (tt7), and also implies the truth part by bivalence of LT . Since L has properties (ii), then for every predicate P ∈ Pm with arity m ≥ 1 the following properties hold in the valuation of L, and hence in the valuation (I) by Lemma 4.2. (p6) The sentence of the form q1x1 . . . qmxmP (x1, . . . , xm) is true iff P (b1, . . . , bm) is true for all bi ∈ N i P when qi is ∀, and for some bi ∈ N i P when qi is ∃. q1x1 . . . qmxmP (x1, . . . , xm) is false iff P (b1, . . . , bm) is false for all bi ∈ N i P when qi is ∃, and for some bi ∈ N i P when qi is ∀. If P is a predicate of P with arity m ≥ 1, and if q1, . . . , qm is any m-tuple of quantifiers ∀ and ∃, then the sentence q1x1 . . . qmxmT (⌈P (ẋ1, . . . , ẋm)⌉) is true iff it is in L0(U) iff it is in Z m 1 iff (by (2.1)) the sentence q1x1 . . . qmxmP (x1, . . . , xm) is true iff (by (p6)) the sentence P (b1, . . . , bm) is true in L, and hence also in LT for all bi ∈ N i P when qi is ∀, and for some bi ∈ N i P when qi is ∃ iff (by (T)) the sentence T (⌈P (b1, . . . , bm)⌉) is true for all choices of bi ∈ N i P when qi is ∀, and for some choices of bi ∈ N i P when qi is ∃. The above equivalences and the bivalence of LT imply the following result. (tp6) The sentence q1x1 . . . qmxmT (⌈P (ẋ1, . . . , ẋm)⌉) true iff the sentence T (⌈P (b1, . . . , bm)⌉) is true for all choices of bi ∈ N i P when qi is ∀, and for some choices of bi ∈ N i P when qi is ∃. q1x1 . . . qmxmT (⌈P (ẋ1, . . . , ẋm)⌉) is false iff the sentence T (⌈P (b1, . . . , bm)⌉) is false for all bi ∈ N i P when qi is ∃, and for some bi ∈ N i P when qi is ∀. Similarly, applying (2.1), (T), (p6) with P replaced by ¬P and bivalence of LT it one can show that T can be replaced in (tp6) by ¬T when P is in Pm and q1, . . . , qm is any m-tuple of quantifiers ∀ and ∃. The above proof shows that LT has properties (ii). 5 Theories of truth Now we are ready to present our main results. Theorem 5.1. Let L0 be a countable language which conforms to classical logic and has not a truth predicate. The language LT defined in Definition 4.1 and valuated by (I) has properties (i)–(iii), and has a definitional theory of truth (shortly DTT). T is a truth predicate, and ¬T is a non-truth predicate. Proof. Properties (i)–(iii) given in Introduction are valid for LT by Lemma 4.2 and Lemma 4.3. Thus LT conforms to classical logic. The results (T) derived in the proof of Lemma 4.3 and biconditional rule (t5) imply that the sentence T (⌈A⌉) ↔ A is true and the sentence ¬T (⌈A⌉) ↔ A is false for every sentence A of LT . T and ¬T are predicates of LT , and their domain DT , the set all sentences of LT , satisfies the condition presented in [3, p. 7] for the domains of truth predicates. Consequently, T is a truth predicate and ¬T is a non-truth predicate. The above results imply that LT has a theory of truth. It is definitional, since truth values of sentences are defined by (I). 10 Next we show that LT has a semantical theory of truth under the following assumptions. (s1) The object language L0 is countable, has not a truth predicate, and every sentence of L0 is meaningful and is valuated by its meaning either as true or as false. (s2) Standard meanings are assigned to logical symbols. (s3) The sentence T (n) means: 'the sentence whose Gödel number has n as its numeral is true'. At first we prove preliminary Lemmas. Lemma 5.1. Under the hypotheses (s1) and (s2) the object language L0 has an extension L which has properties (i)–(iii) given in Introduction when its sentences are valuated by their meanings. In particular, L0 conforms to classical logic. Proof. The object language L0 is bivalent by (s1). The basic extension L of L0 which has properties (i)– (iii) when its sentences are valuated by their meanings is constructed as follows. The first extension L1 of L0 is formed by adding those sentences ¬A, A∨B, A∧B, A → B, A ↔ B and q1x1 . . . qmxmP (x1, . . . , xm) which are not in L0 when A and B go through all sentences of L0, P its predicates and their negations (added if necessary), and (q1, . . . , qm) m-tuples, where qi's are either ∀ or ∃. If there exist objects of D without names, add terms to name them. Valuating the sentences of L1 by their meanings, the assumptions (s1) and (s2) ensure that properties (ii) and (iii) are valid. When the language Ln, n ≥ 1 is defined, define the language Ln+1 by adding those sentences ¬A, A∨B, A∧B, A → B, A ↔ B which are not in Ln when A and B are sentences of Ln. Valuating the sentences of Ln+1 by their meanings, the properties (ii) and (iii) are valid by assumptions (s1) and (s2). The union L of languages Ln, n ∈ N0 has also properties (ii) and (iii). If A and B denote sentences of L, there exist n1 and n2 in N0 such that A is in Ln1 and B is in Ln2 . Denoting n = max{n1, n2}, then A and B are sentences of Ln. Thus the sentences ¬A, A ∨ B, A ∧ B, A → B and A ↔ B are in Ln+1, and hence in L. If P is a predicate of L1 with arity m ≥ 1, then for each m-tuple (q1, . . . , qm), where qi's are either ∀ or ∃, the sentence q1x1 . . . qmxmP (x1, . . . , xm) is in L1, so that it is in L. Thus L has also properties (i). Consequently, the basic extension L of L0 constructed above has all properties (i) – (iii) when its sentences are valuated by their meanings. This implies that L0 conforms to classical logic. Lemma 5.2. Make the assumptions (s1)–(s3), and let L be the language constructed in the proof of Lemma 5.1. Let LT , D and U be as in Definition 4.1, and let W be the set of Gödel numbers of true sentences of L. Given a consistent subset V of D which satisfies W ⊆ V ⊆ U , assume that every sentence of LT whose Gödel number is in V is true and not false by its meaning. Then every sentence of L(V ) (defined by (2.6) with U replaced by V ) is true and not false by its meaning. Proof. Because V ⊆ U = G(U), then every sentence whose Gödel number is in V , is in LT . At first we prove that every sentence of L0(V ) is true and not false by its meaning. By Lemma 5.1 L is bivalent. Thus every true sentence of L, i.e., every sentence of Z is true and not false by its meaning. Let A denote a sentence of LT . By (s3) the sentence T (⌈A⌉) means that 'the sentence whose Gödel number has ⌈A⌉ as its numeral, i.e., the sentence A, is true'. Thus, by its meaning, T (⌈A⌉) is true iff A is true and false iff A is false. Assume that the Gödel number of a sentence A is in V . Since A is by a hypothesis true and not false by its meaning, then T (⌈A⌉) is true and not false by its meaning. This implies by (2.2) that the sentences 11 of Z1(V ) are true and not false by their meanings. By the standard meaning of negation the sentence ¬T (⌈A⌉) is false and not true by its meaning. Replacing A by T (⌈A⌉) and ¬T (⌈A⌉) it follows from the above results that the sentences T (⌈T (⌈A⌉)⌉) and ¬T (⌈¬T (⌈A⌉)⌉) are true and not false by their meanings, and the sentences T (⌈¬T (⌈A⌉)⌉) and ¬T (⌈T (⌈A⌉)⌉) are false and not true by their meanings. Let A denote such a sentence of LT , that the Gödel number of the sentence ¬A is in V . ¬A is by a hypothesis true and not false by its meaning, so that A is false and not true by its meaning since V is consistent. Thus the sentence T (⌈A⌉) is false and not true by its meaning, and the sentence ¬T (⌈A⌉) is true and not false by its meaning. It then follows from (2.2) that the sentences of Z2(V ) are true and not false by their meanings. Replacing A by T (⌈A⌉) and ¬T (⌈A⌉) we then obtain that the sentences T (⌈T (⌈A⌉)⌉) and ¬T (⌈¬T (⌈A⌉)⌉) are false and not true by their meanings, and the sentences ¬T (⌈T (⌈A⌉)⌉) and T (⌈¬T (⌈A⌉)⌉) are true and not false by their meanings. The set NT of numerals, defined by (4.2), is formed by numerals ⌈A⌉, where A goes through all the sentences of LT . Thus, by results proved above T (n), T (⌈T (n)⌉), ¬T (⌈¬T (n)⌉), ¬T (n), ¬T (⌈T (n)⌉) and T (⌈¬T (n)⌉) are for some n ∈ NT true and not false by their meanings and for some n ∈ NT false and not true by their meanings. These results and the standard meanings of quantifiers and negation imply that ∃xT (x), ∃xT (⌈T (ẋ)⌉), ∃x¬T (⌈¬T (ẋ)⌉), ∃x¬T (x), ∃x¬T (⌈T (ẋ)⌉) and ∃xT (⌈¬T (ẋ)⌉) are true and not false by their meanings, and their negations are false and not true by their meanings, whereas ∀xT (x), ∀xT (⌈T (ẋ)⌉), ∀x¬T (⌈¬T (ẋ)⌉), ∀x¬T (x), ∀x¬T (⌈T (ẋ)⌉) and ∀xT (⌈¬T (ẋ)⌉) are false and not true by their meanings, and their negations are true and not false by their meanings. By above results and (2.2) the sentences of Z1 and Z2 are true and not false by their meanings. Let P be a predicate in Pm, and let q1, . . . , qm be an m-tuple of quantifiers ∀ and ∃. Since the sentences q1x1 . . . qmxmP (x1, . . . , xm) and P (b1, . . . , bm) are in L, they are by their meanings either true and not false, or false and not true. If q1x1 . . . qmxmP (x1, . . . , xm) is true and not false by its meaning, then P (b1, . . . , bm) is true and not false by its meaning for all bi when qi is ∀, and for some bi when pi is ∃. Thus, by (s3), T (⌈P (b1, . . . , bm)⌉) is true and not false by its meaning for all bi when qi is ∀, and for some bi when pi is ∃. Consequently, q1x1 . . . qmxmT (⌈P (b1, . . . , bm)⌉) is true and not false by its meaning. By its meaning ¬P (b1, . . . , bm) is true and not false iff P (b1, . . . , bm) is false and not true iff, by (s3), T (⌈P (b1, . . . , bm)⌉) is false and not true iff ¬T (⌈P (b1, . . . , bm)⌉) is true and not false. Thus the sentence q1x1 . . . qmxm¬T (⌈P (ẋ1, . . . , ẋm)⌉) is true and not false by its meaning iff the sentence q1x1 . . . qmxm¬P (x1, . . . , xm) is true and not false by its meaning. It follows from the above results and (2.1) that the sentences of Zm1 and Z m 2 are true and not false by their meanings. Consequently, the sentences of Z3 and Z4 are by (2.2) true and not false by their meanings. The above results and (2.3) imply that every sentence of L0(V ) is true and not false by its meaning. Thus the following property holds when n = 0. (h3) Every sentence of Ln(V ) is true and not false by its meaning. Make the induction hypothesis: (h3) holds for some n ∈ N0. Given a sentence of L0n(V ), it is of the form ¬(¬A), where A is in Ln(V ). A is by (h3) true and not false by its meaning. Thus, by standard meaning of negation, its double application implies that the sentence ¬(¬A), and hence the given sentence, is true and not false by its meaning. If a sentence is in L1n(V ), it is of the form A ∨ B, where A or B is in Ln(V ). By (h3) at least one of the sentences A and B is true and not false by its meaning. Thus the sentence A∨B, and hence given sentence, is true and not false by its meaning. 12 Similarly it can be shown that if (h3) holds, then every sentence of Lkn(V ), where 2 ≤ k ≤ 8, is true and not false by its meaning. The above results imply that under the induction hypothesis (h3) every sentence of Lkn(V ), where 0 ≤ k ≤ 8, is true and not false by its meaning. It then follows from the definition (2.5) of Ln+1(V ) that if (h3) is valid for some n ∈ N0, then every sentence of Ln+1(V ) is true and not false by its meaning. The first part of this proof shows that (h3) is valid when n = 0. Thus, by induction, it is valid for all n ∈ N0. This result and (2.6) imply that every sentence of L(V ) is true and not false by its meaning. Lemma 5.3. Let U be the smallest consistent subset of D which satisfies U = G(U). Then under the hypotheses (s1)–(s3) every sentence of LT whose Gödel number is in U is true and not false by its meaning. Proof. By Theorem 3.1 the smallest consistent subset U of D which satisfies U = G(U) is the last member of the transfinite sequence (Uλ)λ<γ constructed in the proof of that Theorem. We prove by transfinite induction that the following result holds for all λ < γ. (H) Every sentence of LT whose Gödel number is in Uλ is true and not false by its meaning. Make the induction hypothesis: There is a μ satisfying 0 < μ < γ such that (H) holds for all λ < μ. Let λ < μ be given. Because Uλ is consistent and W ⊆ Uλ ⊆ U for every λ < μ, it follows from the induction hypothesis and Lemma 5.2 that every sentence of L(Uλ) is true and not false by its meaning. This implies by (3.1) that (H) holds when Uλ is replaced G(Uλ), for every λ < μ. Thus (H) holds when Uλ is replaced by the union of those sets. But this union is Uμ by Theorem 3.1 (G), whence (H) holds when λ = μ. When μ = 1, then λ < μ iff λ = 0. U0 = W , i.e., the set of Gödel numbers of true sentences of L. Since L, valuated by meanings of its sentences, is bivalent by Lemma 5.1, the sentences of U0 are true and not false by their meanings. This proves that the induction hypothesis is satisfied when μ = 1. The above proof implies by transfinite induction properties assumed in (H) for Uλ whenever λ < γ. In particular the last member of (Uλ)λ<γ satisfies (H), which is by Theorem 3.1 the smallest consistent subset U of D for which U = G(U). This proves the assertion. The next result is a consequence of Lemma 5.1, Lemma 5.3 and Theorem 5.1. Theorem 5.2. Under the hypotheses (s1)–(s3) the extension LT of L0 defined in Definition 4.1 has a semantical theory of truth (shortly STT), when the valuation (I) is replaced in Theorem 5.1 with the valuation of the sentences of LT by their meanings. This valuation is equivalent to valuation (I), and the results of Theorem 5.1 are valid for STT. Proof. Let A denote a sentence of LT . A is by Definition 4.1 in L(U) or in F (U), where U is the smallest consistent subset of D which satisfies U = G(U). If A is in L(U), its Gödel number is in G(U) = U by (3.1), whence it is by Lemma 5.3 true and not false by its meaning. If A is in F (U), then ¬A is in L(U) by (4.1). Thus ¬A is true and not false by its meaning, so that A is by the standard meaning of negation false and not true by its meaning. Hence the valuation of LT by meanings of its sentences is equivalent to valuation (I). In particular, LT has properties (i)-(iii). Moreover, the sentence T (⌈A⌉) ↔ A is true by its meaning and the sentence ¬T (⌈A⌉) ↔ A is false by its meaning for every sentence A of LT . These results imply that T is a truth predicate and ¬T is a non-truth predicate for LT . Consequently, LT has a semantical theory of truth. 13 6 Compositionality of truth in theories DTT and STT One of the desiderata introduced in [7] for theories of truth is that truth should be compositional. In this section we present some logical equivalences which theories DTT and STT of truth prove. Lemma 6.1. Theories DTT and STT of truth presented in Theorems 5.1 and 5.2 prove the following logical equivalences when A and B are sentences of LT , and P is in P or P is T . (a0) T (⌈. . . ⌈T (⌈A⌉)⌉ . . . ⌉) ↔ T (⌈T (⌈A⌉)⌉) ↔ T (⌈A⌉) ↔ A. (a1) ¬T (⌈A⌉) ↔ T (⌈¬A⌉) ↔ ¬A. (a2) T (⌈A⌉) ∨ T (⌈B⌉) ↔ T (⌈A ∨ B⌉) ↔ A ∨ B. (a3) T (⌈A⌉) ∧ T (⌈B⌉) ↔ T (⌈A ∧ B⌉) ↔ A ∧ B. (a4) (T (⌈A⌉) → T (⌈B⌉)) ↔ T (⌈A → B⌉) ↔ (A → B). (a5) (T (⌈A⌉) ↔ T (⌈B⌉)) ↔ T (⌈A ↔ B⌉) ↔ (A ↔ B). (a6) ¬T (⌈A ∨ B⌉) ↔ ¬(A ∨ B) ↔ ¬A ∧ ¬B ↔ T (⌈¬A⌉) ∧ T (⌈¬B⌉) ↔ ¬T (⌈A⌉) ∧ ¬T (⌈B⌉). (a7) ¬T (⌈A ∧ B⌉) ↔ ¬(A ∧ B) ↔ ¬A ∨ ¬B ↔ T (⌈¬A⌉) ∨ T (⌈¬B⌉) ↔ ¬T (⌈A⌉) ∨ ¬T (⌈B⌉). (a8) ∀xT (⌈P (ẋ)⌉) ↔ T (⌈∀xP (x)⌉) ↔ ∀xP (x) ↔ ¬∃x¬P (x) ↔ T (⌈¬∃x¬P (x)⌉) ↔ ¬∃x¬T (⌈P (ẋ)⌉). (a9) ∃xT (⌈P (ẋ)⌉) ↔ T (⌈∃xP (x)⌉) ↔ ∃xP (x) ↔ ¬∀x¬P (x) ↔ T (⌈¬∀x¬P (x)⌉) ↔ ¬∀x¬T (⌈P (ẋ)⌉). (a10) ¬T (⌈∀xP (x)⌉) ↔ T (⌈¬∀x(P (x)⌉) ↔ ¬∀xP (x) ↔ ∃x¬P (x) ↔ T (⌈∃x¬P (x)⌉). (a11) ¬T (⌈∃xP (x)⌉) ↔ T (⌈¬∃xP (x)⌉) ↔ ¬∃xP (x) ↔ ∀x¬P (x) ↔ T (⌈∀x¬P (x)⌉). Proof. T-rule implies equivalences of (a0). The first equivalences in (a1)–(a5) are easy consequences of rules (t1)–(t5) and T -rule (cf. [5, Lemma 4.1]). Their second equivalences are consequences of T -rule. The first and last equivalences of (a6) and (a7) are consequences of (a1). Their second equivalences are De Morgan's laws of classical logic (cf. [1]). Third equivalences of (a6) and (a7) follow from T -rule. The first equivalences of (a8) and (a9) are easy consequences of rules (tp6) and (tp7) and T -rule (cf. [5, Lemma 4.2]). T -rule implies their second equivalences. The third equivalences are De Morgan's laws for quantifiers (cf. [1]). The fourth ones follow from T -rule. De Morgan's laws with P (x) replaced by T (⌈P (ẋ)⌉) imply equivalence of the last and the first sentences. (a10) and (a11) are negations to some equivalences of (a8) and (a9). Let L0 be a countable and bivalent first-order language with or without identity. It conforms to classical logic. If P and Q are predicates of L0 with arity 1 and domain D, then ¬P , P ∨ Q, P ∧ Q, P → Q and P ↔ Q are predicates of L0 with domain D. Replacing P and/or Q by some of them we obtain new predicates with domain D, and so on. Thus P in (a8) and (a9) can be replaced by anyone of these predicates. Their universal and existential quantifications are sentences of L. They are also sentences of LT . Anyone of them can be the sentence A and/or the sentence B in results (a1)–(a7) derived above. Moreover, P can be replaced by anyone of those predicates in (a8)–(a11). Take a few examples. ∀xT (⌈P (ẋ) → Q(ẋ)⌉) ↔ T (⌈∀x(P (x) → Q(x))⌉) ↔ ∀x(P (x) → Q(x)). ∃xT (⌈P (ẋ) ∧Q(ẋ)⌉) ↔ T (⌈∃x(P (x) ∧Q(x)⌉) ↔ ∃x(P (x) ∧Q(x)). ∀xT (⌈P (ẋ) → ¬Q(ẋ)⌉) ↔ T (⌈∀x(P (x) → ¬Q(x))⌉) ↔ ∀x(P (x) → ¬Q(x)). ∃xT (⌈P (ẋ) ∧ ¬Q(ẋ)⌉) ↔ T (⌈∃x(P (x) ∧ ¬Q(x)⌉) ↔ ∃x(P (x) ∧ ¬Q(x)). 14 These equivalences correspond to the four Aristotelian forms: 'All P 's are Q's', 'some P 's are Q's', 'no P 's are Q's' and 'some P 's are not Q's' (cf. [1]). Let P be a predicate of L0 with arity m > 1, and let q1, . . . qm be any of the 2 m different m-tuples which can be formed from quantifiers ∀ and ∃. Theories DTT and STT presented in Theorems 5.1 and 5.2 prove the following logical equivalences. T (⌈q1x1 . . . qmxmP (x1, . . . , xm)⌉) ↔ q1x1 . . . qmxmP (x1, . . . , xm) ↔ q1x1 . . . qmxmT (⌈P ((ẋ1, . . . , ẋm)⌉), T (⌈q1x1 . . . qmxm¬P (x1, . . . , xm)⌉) ↔ q1x1 . . . qmxm¬P (x1, . . . , xm) ↔ q1x1 . . . qmxm¬T (⌈P (ẋ1, . . . , ẋm)⌉). T -rule implies the first equivalences, and the second equivalences are consequences of (2.1) and bivalence of LT . An application of T -rule proves the universal T -schema: (UT) ∀x1 . . .∀xm ( T (⌈P ((ẋ1, . . . , ẋm)⌉) ↔ P (x1, . . . , xm) ) . Example 6.1. Assume that L0 is the language of arithmetic with its standard interpretation. Let R(x, y) be formula x = 2y, and let R be the corresponding predicate with domain DR = N0 × N0. Then the truth theories DTT and STT of the extension LT of L0 prove the universal T -schema (UTR) ∀x∀y ( T (⌈R((ẋ, ẏ)⌉) ↔ R(x, y) ) , and the logical equivalences (q1) q1xq2yT (⌈R(ẋ, ẏ)⌉) ↔ T (⌈q1xq2yR(x, y)⌉) ↔ q1xq2yR(x, y), (q2) q1xq2y¬T (⌈R(ẋ, ẏ)⌉) ↔ ¬T (⌈q1xq2yR(x, y)⌉) ↔ q1xq2y¬R(x, y). The sentences in (q1) are true iff q1q2 is ∀∃ or ∃∃, and false iff q1q2 is ∀∀ or ∃∀. In (q2) the sentences are true iff q1q2 is ∀∀ or ∀∃, and false iff q1q2 is ∃∀ or ∃∃. The next example is a modification of an example presented in [10, p. 704]. Example 6.2. Let L0, LT , T and R(x, y) be as in Example 6.1. Denote by T ′ the predicate with domain N0 corresponding to formula ∃yR(x, y). Let the sublanguage L ′ 0 of L0 be formed by the syntax of L0, predicate T ′ and sentences ∃yR(n, y), where n goes through all numerals, the names of natural numbers n. Choose n to be the code number of the sentence ∃yR(n, y) in L′0 for each numeral n. If A is a sentence of L′0, i.e. ∃yR(n, y) for some numeral n, then n is the code number of A, so that ⌈A⌉ is n. Since T ′(n) denotes the sentence ∃yR(n, y) for each numeral n, then T ′(⌈A⌉) or equivalently T ′(n), is true (respectively false) in L′0 iff ∃yR(n, y), or equivalently A, is true, (respectively false) in L ′ 0 iff (by definition of R(x, y)) n is even (respectively odd). Thus T ′ is a truth predicate of L′0. The restriction of T to L′0 is also a truth predicate of L ′ 0. It is not equal to T ′ because the coding of L′0 is not the restriction of the Gödel numbering of LT , whereas L0 can have that syntax. 7 Remarks Theories DTT and STT conform to the seven desiderata presented in [7] for theories of truth (cf. [5, Theorem 4.2]). This challenges the current view (cf, e.g., [3, 7, 9]). Theories DTT and STT are also free from paradoxes. The fact that the languages LT which have these theories of truth contain their own truth predicates seems to be in contrast to some Tarski's limiting theorems for theories of truth described, e.g., in [10]. Conformity of theories DTT and STT to the principles of classical logic can be vitiated by adding first-order syntax to LT . The object language L0 is allowed to have that syntax. Theorems 5.1 and 5.2 imply that theories DTT and STT of truth together contain the theory DSTT of truth introduced in [5, Theorem 4.1] for languages L0 when those sentences of the form A∨B, A → B and ¬(A∧B) whose one component has not a true value are deleted from L0. Otherwise the languages LT for which theories DTT and STT of truth are introduced extend languages L 0. The amount of predicates and compositional sentences are multiplied by means of the added non-truth predicate ¬T , and predicates of the object language L0 which have several free variables. 15 The family of those languages which conform to classical logic is considerably larger than the families of those object languages considered in [5]. For instance, the object language L0 can be any language whose every sentence is valuated by its meaning either as true or as false. Every language L0 which has properties (i) – (iii), e.g., every countable and bivalent first-order language with or without identity, conforms to classical logic. If L0 is any countable language conforming to classical logic, and if L ′ 0 is any sublanguage of L0 formed by the syntax of L0, any nonempty subset of its sentences and any subset of its predicates, then L′0 conforms to classical logic. For instance, the language L ′ 0 in Example 6.2 conforms to classical logic. The set U used in the definition of LT is the smallest consistent set for which U = G(U), where G(U) is the set of Gödel numbers of sentences of L(U). Thus U is the minimal fixed point of the mapping G : C → C, where C is the set of consistent sets of Gödel numbers of sentences of L. Moreover LT conforms to classical logic. Thus the sentences of LT are grounded in the sense defined by Kripke in [6, p. 18]. The language Lσ determined by the minimal fixed point in Kripke's construction contains also sentences which don't have truth values. For instance, the sentence A ↔ T (⌈A⌉) has not a truth value for every sentence A of Lσ. Thus a three-valued logic is needed in [6], as well as in [3] and in [4]. The only logic used here is classical. The equivalence of truth values of sentences in theories DTT and STT show that the notion of 'grounded truth' defined by valuation (I) conforms to the 'ordinary' notion of truth. In the metalanguage used in the above presentation some concepts dealing with predicates and their domains are revised from those used in [5] so that they agree better with the corresponding concepts in informal languages of first-order logic (cf. [1]). The circular reasoning used in [5] to show that G(U) is consistent if U is consistent is corrected in the proof of Lemma 3.1. Mathematics, especially ZF set theory, plays a crucial role in this paper. 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A Set-Theoretic Explanation, Synthese, https://doi.org/10.1007/s11229-018-1873-x. [9] Raatikainen, Panu (2019) Truth and Theories of Truth, (penultimate draft) The Cambridge Handbook of Philosophy of Language, Cambridge University Press, to appear. [10] Ray, Greg (2018) Tarski on the Concept of Truth, The Oxford Handbook of Truth, Michael Glanzberg ed. Oxford: Oxford University Press, 695–717.