A consistent theory of truth for languages which conform to classical logic

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1 Nonlinear Studies - www. nonlinearstudies.com MESA - Preprint submitted to Nonlinear Studies / MESA A consistent theory of truth for languages which conform to classical logic S. Heikkilä Department of Mathematical Sciences, University of Oulu BOX 3000, FIN-90014, Oulu, Finland Corresponding Author. sheikki@cc.oulu.fi Abstract. For languages which conform to classical logic such extensions are constructed that they conform to classical logic, and possess a consistent theory of truth. Every language, whose sentences have meanings which make them either true or false, is shown to conform to classical logic. 1 Introduction Based on Chomsky Definition (cf. [2]) we assume that a language is a countable set of sentences with finite length, and formed by a countable set of elements. A theory of syntax is also assumed to provide symbols for a language and rules to construct well-formed sentences and possible formulas. A language is said to conform to classical logic, if it has, or if it can be extended to have the following properties: (i) It contains logical symbols (not), (or), (and), (if...then), (if and only if), (for all) and (exists), and the following sentences: If A and B are (denote) sentences, so are A, A B, A B, A B and A B. If P(x) is a formula of the language, then P is called a predicate with domain X P if P(b) is a sentence of that language whenever b is a term which names an object of X P. Denote by N (X P ) the set of those terms. xp(x) and xp(x) are then sentences of the language. If P has several free variables x 1,...,x m, then P(x 1,...,x m ) is denoted by P(x), and the sentences xp(x) and xp(x) stand for universal and existential closures x 1 x m P(x 1,...,x m ) and x 1 x m P(x 1,...,x m ). (ii) The sentences of that language are so interpreted that the following rules of classical logic hold ( iff means if and only if ): 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 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 domain X P, then xp(x) is true 2010 Mathematics Subject Classification: 03B10, 03B65, 03D80, 91F20, 97M80 Keywords: language, sentence, interpretation, meaning, truth predicate, consistent, logic, classical

2 2 iff P(b) is true for every b N (X P ), and false iff P(b) is false for some b N (X P ); xp(x) is true iff P(b) is true for some b N (X P ), and false iff P(b) is false for every b N (X P ). (iii) The language is bivalent: Every sentence of it is interpreted either as true or as false. Main results of this paper are: Every language which conforms to classical logic has an extension which has properties (i) (iii), and for which a consistent theory of truth is formulated. Every language, whose sentences have meanings which make them either true or false, conforms to classical logic. 2 An extended language and its properties Let L 0 be a language which conforms to classical logic. Property (iii) is then necessarily valid, and the interpretation of sentences of L 0 has to obey rules of (ii). The basic extension L of L 0 is constructed as follows: Sentences A, A B, A B, A B, A B, xp(x) and xp(x), where A and B go through all sentences of L 0 and P its predicates, are added if they are not in L 0, and interpreted so that properties (ii) hold. Denote by L 1 the so extended language. It has properties (ii) and (iii). Replacing L 0 by L 1 and so on, we obtain a sequence of languages L n, n N 0 = {0,1,2,...}, whose sentences are so interpreted that the properties (ii) and (iii) are valid. This holds also for the language L which is union of languages L n, n N 0. If A and B denote sentences of L, there exist n 1 and n 2 such that A is in L n1 and B is in L n2. Denoting n = max{n 1,n 2 }, then A and B are sentences of L n. Thus the sentences A, A B, A B, A B and A B are in L n+1, and hence in L. If P is a predicate of L 0, then the sentences xp(x) and xp(x) are in L 1, and hence in L. Thus L has also the syntactic properties (i) so that it has properties (i) (iii). To form an extension of L 0 which has a truth predicate, construct a language L 0 as follows: Its base language is formed by L, an extra formula T (x) and sentences T (n), where n goes through all numerals of numbers n N 0. Numerals are also added, if necessary, to terms of L. Fix a Gödel numbering to the base language. 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 domain X P, then P(b) is a sentence of L for each b N (X P ), and P(b) is the numeral of its Gödel number. Thus T ( P(b) ) is a sentence of L 0 for each b N (X P ), whence T ( P( ) ) is a predicate of L 0 with domain X P. The construction of L 0 is completed by adding to it sentences xt (x), xt (x), xt ( T (x) ) and xt ( T (x) ), and sentences xt ( P(x) ) and xt ( P(x) ) for every predicate P of L. When a language L n, n N 0, is defined, let L n+1 be a language which is formed by adding to L n those of the following sentences which are not in L n : A, A B, A B, A B and A B, where A and B go through all sentences of L n. The language L is defined as the union of languages L n, n N 0. Extend the Gödel numbering of the base language to L, and denote by D the set of those Gödel numbers. Denote by P the set of all predicates of L. Divide P into three disjoint subsets. P 1 = {P P : P(b) is a true sentence of L for every b N (X P )}, P 2 = {P P : P(b) is a false sentence of L for every b N (X P )}, P 3 = {P P : P(b) is a true sentence of L for some but not for all b N (X P )}. (2.1)

3 Consistent theory of truth 3 Define subsets C 1 (U), C 2 (U), U D, and C i, i = 1...4, of L by C 1 (U) = {T (n): n = A, where A is a sentence of L and #A is in U}, C 2 (U) = { T (n): n = A, where A is a sentence of L and #[ A] is in U}, C 1 = { xt (x), xt (x), ( xt ( T (x) )), xt ( T (x) )}, C 2 = { xt ( P(x) ), xt ( P(x) ) : P P 1 }, C 3 = { ( xt ( P(x) )), ( xt ( P(x) )) : P P 2 }, C 4 = { ( xt ( P(x) )), xt ( P(x) ) : P P 3 }. Subsets L n (U), n N 0, of L are defined recursively as follows. { E = {A : A is a true sentence of L} if U = /0 (the empty set), L 0 (U) = E C 1 (U) C 2 (U) C 1 C 2 C 3 C 4 if /0 U D. When n N 0, and a subset L n (U) of L is defined, define Ln(U) 0 = { ( A) : A is in L n (U)}, L 1 n(u) = {A B : A or B is in L n (U)}, L 2 n(u) = {A B : A and B are in L n (U)}, Ln(U) 3 = {A B : A or B is in L n (U)}, Ln(U) 4 = {A B : both A and B or both A and B are in L n (U)}, L 5 n(u) = { (A B) : A and B are in L n (U)}, L 6 n(u) = { (A B) : A or B is in L n (U)}, Ln(U) 7 = { (A B) : A and B are in L n (U)}, Ln(U) 8 = { (A B) : A and B, or A and B are in L n (U)}, (2.2) (2.3) (2.4) and L n+1 (U) = L n (U) 8 Ln(U). k (2.5) The above constructions imply that L n (U) L n+1 (U) L and L k n(u) L k n+1 (U) for all n N 0 and k = 0,...,8. Define subsets L(U) and Q(U) of L by L(U) = k=0 L n (U) and Q(U) = {A : A L(U)}. (2.6) n=0 The subsets G(U) and F(U) of the set D of Gödel numbers of sentences of L, defined by G(U) = {#A : A L(U)} and F(U) = {#A : A L(U)}, (2.7) coincide to those defined in [4]. The properties derived in [4] for G(U) and F(U) are thus available. A subset U of D is said to be consistent if there is no sentence A in L such that both #A and #[ A] are in U. The existence of the smallest consistent subset U of D which satisfies U = G(U) is proved in [4, Theorem 6.1] by a transfinite recursion method. Definition 2.1. Let U be the smallest consistent subset of D which satisfies U = G(U). Denote by L 0 a language which is formed by symbols of L 0 and all the sentences and formulas of L(U) and Q(U).

4 4 (I) A theory of syntax for L 0 consists of that of L 0, symbols of L 0 and rules to construct those sentences and formulas of L 0 which are not in L 0. An interpretation of L 0 is defined as follows. (II) A sentence of L 0 is interpreted as true iff it is in L(U), and as false iff it is in Q(U). Lemma 2.1. Let L 0 be defined by Definition 2.1 and interpreted by (II). Then a sentence of the basic extension L of L 0 is true (respectively false) in the interpretation of L iff it is true (respectively false) in the interpretation (II). Proof. Let A denote a sentence of L. A is true in the interpretation (II) iff A is in L(U) iff (by the construction of L(U)) A is in E iff A is true in the interpretation of L. A is false in the interpretation (II) iff A is in Q(U) iff (by (2.6)) A is in L(U) iff ( A is a sentence of L) A is in E iff A is true in the interpretation of L iff (L conforms to classical logic) A is false in the interpretation of L. Proposition 2.1. Let L 0 be a language which conforms to classical logic, and is without a truth predicate. Then the language L 0 defined by Definition 2.1 and interpreted by (II) has properties (i) (iii) given in Introduction. T is a predicate of L 0 when its domain X T and N (X T ) are defined by X T = {the sentences of L 0 } and N (X T ) = {n : n = A, where A is a sentence of L 0 }. (2.8) Proof. It follows from the construction of L 0 that it has properties (i). To show that (iii) holds, let A denote a sentence of L 0. Then #A is in G(U) or in F(U). Because U is consistent and U = G(U), then G(U) and F(U) are disjoint by [4, Lemma 3.2]. Thus #A is either in G(U) or in F(U), so that A is by (2.7) either in L(U) or in Q(U), and hence either true or false in the interpretation (II). Consequently, L 0 is bivalent, which proves (iii). Next we shall show that the interpretation rules given in (ii) are satisfied. If A and B denote arbitrary sentences of L 0, then the following rules should hold. (t1) Negation: A is true iff A is false, and A is false iff A is true. (t2) Disjunction: A B is true iff A or B is true. A B false iff A and B are false. (t3) Conjunction: A B is true iff A and B are true. A B is false iff A or B is 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. We shall first derive the following auxiliary rule. (t0) Double negation: If A is a sentence of L 0, then ( A) is true iff A is true. To prove (t0), assume first that ( A) is true. Then ( A) is in L(U). By (2.6) ( A) is in L n (U) for some n N 0. If ( A) is in L 0 (U) then ( A) is in E, so that sentence ( A) is true in the interpretation of L. This holds (negation rule is valid in L) iff A is false in the interpretation of L iff A is true in the interpretation of L. Thus A is in E L(U), whence A is true. Assume next that the least of those n for which ( A) is in L n (U) is > 0. The definition of L n (U) implies that if ( A) is in L n (U), then ( A) is in L 0 n 1 (U), so that A is in L n 1(U), and hence in L(U), i.e., A is true. Thus A is true if ( A) is true.

5 Consistent theory of truth 5 Conversely, assume that A is true. Then A is in L(U), so that A is in L n (U) for some n N 0. Thus ( A) is in L 0 n(u), and hence in L n+1 (U). Consequently, ( A) is in L(U), whence ( A) is true. This concludes the proof of (t0). Rule (t0) is applied to prove negation rule (t1). Let A be a sentence of L 0. Then A is true iff (by (t0)) ( A) is true iff ( A) is in L(U) iff (by (2.6)) A is in Q(U) iff A is false. A is false iff A is in Q(U) iff (by (2.6)) A is in L(U) iff A is true. Thus (t1) is satisfied. Next we shall prove rule (t3). Let A and B be sentences of L 0. If A and B are true, i.e., A and B are in L(U), there is by (2.6) an n N 0 such that A and B are in L n (U). Thus A B is 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 N 0 such that A B is in L n (U). If A B is in L 0 (U), it is in E. Thus A B is true in the interpretation of L. Because rule (t3) is valid in L, then A and B are true in the interpretation of L, and hence also in the interpretation (II) by Lemma 2.1. Assume next that the least of those n for which A B is in L n (U) is > 0. Then A B is in L 2 n 1 (U), so that A and B are in L n 1 (U), and hence in L(U), i.e., A and B are true. Consequently, A B is true iff A and B are true. It follows from (2.6) that (a) A B is in Q(U) iff (A B) is in L(U). If (A B) is in L(U), there is by (2.6) an n N 0 such that (A B) is in L n (U). If (A B) is in L 0 (U), it is in E. Then (A B) is true and A B is false in the interpretation of L. Thus A or B is false so that A or B is true in the interpretation of L, i.e., A or B is in E, and hence in L(U). Assume next that the least of those n for which (A B) is in L n (U) is > 0. Then (A B) is in L 6 n 1 (U), so that A or B is in L n 1(U), and hence in L(U). Thus, A or B is in L(U) if (A B) is in L(U). Conversely, if A or B is in L(U), there exist by (2.6) n N 0 such that A or B is in L n (U). This result and the definition of L 6 n(u) imply that (A B) is in L 6 n(u), and hence in L(U). Consequently, (b) (A B) is in L(U) iff A or B is in L(U) iff (by (2.6)) A or B is in Q(U). Thus, by (a) and (b), A B is in Q(U) iff A or B is in Q(U). But this means that A B is false iff A or B is false. This concludes the proof of (t3). The proofs of rules (t2), (t4) and (t5) are similar to the above proof of (t3). Because U = G(U), it follows from (2.2), (2.6) and (2.7) that T (n) is a sentence of L 0 whenever n = A, where A is a sentence of L 0. Thus T is a predicate of L 0 when its domain X T and N (X T ) are defined by (2.8). In order that T has properties required for predicates in (ii), we shall show that the following rules are valid. (t6) xt (x) is true iff T (n) is true for some n N (X T ). xt (x) is false iff T (n) is false for every n N (X T ). (t7) xt (x) is true iff T (n) is true for every n N (X T ). xt (x) is false iff T (n) is false for some n N (X T ). Because U is nonempty, then xt (x) is in L 0 (U) by (2.2) and (2.3), and hence in L(U) by (2.6). Thus xt (x) is by (II) and by bivalence of L 0 true but not false sentence of L 0. If #A 1 is in G(U) = U, then T (n) is by (2.2) in C 1 (U), and hence in L(U) when n = A 1. This result and (II) imply that T (n) is true when n = A 1, and hence for some n N (X T ). Because L 0 is bivalent, then T (n) is not false when n = A 1, and hence not false for every n N (X T ).

6 6 The above proof implies that T satisfies rule (t6). xt (x) is in L 0 (U), and hence in L(U), so that it is true. Thus xt (x) is false but not true by (t1) and (iii). If #[ A 2 ] is in G(U) = U, then T (n) is by (2.2) in C 2 (U) L(U), and hence true when n = A 2. Thus T (n) is by (t1) and (iii) false but not true when n = A 2. Thus T (n) is false for some n N (X T ) and not true for every n N (X T ). As a consequence of the above reasoning we infer that rule (t7) is valid. It remains to show that the following rules hold. (tt6) xt ( T (x) ) is true iff T ( T (n) ) is true for some n N (X T ), and xt ( T (x) ) is false iff T ( T (n) ) is false for every n N (X T ); (tt7) xt ( T (x) ) is true iff T ( T (n) ) is true for every n N (X T ), and xt ( T (x) ) is false iff T ( T (n) ) is false for some n N (X T ); and if P is a predicate of L with domain X P, then (tp6) xt ( P(x) ) is true iff T ( P(b) ) is true for some b N (X P ), and xt ( P(x) ) false iff T ( P(b) ) is false for every b N (X P ); (tp7) xt ( P(x) ) is true iff T ( P(b) ) is true for every b N (X P ), and xt ( P(x) ) false iff T ( P(b) ) is false for some b N (X P ). To begin with properties (tp6) and (tp7), consider first the case when P P 1. Then P(b) is a true sentence of L for every b N (X P ). This implies by property (iii) that P(b) is not false for any b N (X P ). Since U is nonempty and proper subset of D, then xt ( P(x) ) and xt ( P(x) ) are in L 0 (U) by (2.2) and (2.3), and hence in L(U). Thus xt ( P(x) ) and xt ( P(x) ) are by (II) and (iii) true but not false sentences of L 0. Since every true sentence of L is by Lemma 2.1 a true sentence of L 0, then, by the interpretation (II), #P(b) is in G(U) = U for every b N (X P ). Thus T ( P(b) ) is by (2.2) in C 1 (U), and hence in L(U), for every b N (X P ). Consequently, by (II) and (iii), T ( P(b) ) is true but not false for every b N (X P ), and hence also for some b N (X P ). The above results imply that P has properties (tp6) and (tp7) when P is in P 1. The proof in the case when P is in P 2 is similar. Assume next that P is in P 3. Then P(b) is a true sentence of L for some b N (X P ), say b N (XP 1), and a false sentence of L for b N (XP 2) = N (X P) \ N (XP 1 ). Since /0 U D, then xt ( P(x) ) and ( xt ( P(x) )) are in L 0 (U) by (2.2) and (2.3), and hence in L(U) by (2.3) and (2.6). Thus xt ( P(x) ) is by (II) and (iii) true but not false, and xt ( P(x) ) is by (II), (t1) and (iii) false but not true. Since every true sentence of L is by Lemma 2.1 a true sentence of L 0 and every false sentence of L is a false sentence of L 0, then, by the interpretation (II), #P(b) is in G(U) = U for every b N (XP 1), and in F(U) for every b N (XP 2 ). In the latter case #[ P(b)] is in G(U) = U by (2.6) and (2.7). But then, by (2.2), T ( P(b) ) is in C 1 (U), and hence in L(U), for every b N (XP 1 ), and T ( P(b) ) is in C 2 (U), and hence in L(U), for every b N (XP 2 ). Thus, by (II), (iii) and (t1) T ( P(b) ) is true but not false for every b N (XP 1), and T ( P(b) ) is false but not true for every b N (X P 2). Consequently, properties (tp6) and (tp7) hold also also when P is in P 3. The proof of properties (tt6) and (tt7) is similar to that given above for properties (tp6) and (tp7) in the case when P is in P 3. The above results imply that the interpretation rules (ii) given in Introduction are satisfied in the interpretation (II) of L 0. This concludes the proof.

7 Consistent theory of truth 7 3 A consistent theory of truth Now we are ready to prove our main result. Theorem 3.1. Assume that an object language L 0 is without a truth predicate and conforms to classical logic. Then the language L 0 constructed in Section 2 and interpreted by (II) conforms to classical logic. Moreover, A T ( A ) is true and A T ( A ) is false for every sentence A of L 0, and T is a truth predicate for L 0. The so obtained theory of truth for L 0 is consistent. Proof. Let A denote a sentence of L 0. The interpretation (II), rule (t1), the definitions of C 1 (U), C 2 (U), G(U) and F(U), and the assumption U = G(U) in Definition 2.1 imply that A is true iff #A is in G(U) = U iff T ( A ) is in C 1 (U) L(U) iff T ( A ) is true and T ( A ) is false. A is false iff #A is in F(U) iff #[ A] is in G(U) = U iff T ( A ) is in C 2 (U) L(U) iff T ( A ) is true and T ( A ) is false. The above results and biconditional rule (t5) imply that A T ( A ) is true and A T ( A ) is false for every sentence A of L 0. T is by Proposition 2.1 a predicate of L 0, and the domain X T of T is the set all sentences of L 0. Thus X T satisfies the condition presented in [3, p. 7] for the domains of truth predicates. Consequently, T is a truth predicate for L 0. The so obtained theory of truth for L 0 is consistent, i.e., free from contradictions, since L 0 is bivalent. As a consequence of [4, Proposition 3.3] we obtain the following result. Proposition 3.1. Assume that L 0 is a language whose sentences have meanings which make them either true or false, and is without a truth predicate. Then the language L 0 constructed in Section 2 can be interpreted by meanings of its sentences, and has properties (i) (iii) given in Introduction. Moreover, this interpretation is equivalent to that of (II). Proof. Let L 0 be a language whose sentences have meanings which make them either true or false. This bivalence remains valid for sentences of the basic extension L of L 0 constructed in Section 2 when the sentences which contain logical symbols are interpreted by their standard meanings. Thus the object language L 0 has an extension L whose sentences have meanings which make them either true or false, which is without a truth predicate, and which has properties (i) and (ii). These properties are valid also for L 0 when meanings of its sentences are determined by meanings of the sentences of L, by meaning of T ( A ), i.e., The sentence denoted by A is true, and by standard meanings of logical symbols. Since L and L 0 have properties (i) (iii), they are fully interpreted by meanings of their sentences in the sense defined in [4]. Results of [4, Subsection 3.2] are then available, whence the last conclusion follows from [4, Proposition 3.3]. The next result is a consequence of Proposition 3.1 and Theorem 3.1. Proposition 3.2. Every language whose sentences have meanings which make them either true or false has an extension possessing the theory of truth formulated in Theorem 3.1. Results of Theorem 3.1 and Proposition 3.2 imply that the theory DSTT of truth formulated in [4, Theorem 4.1] is valid for languages which conform to classical logic. Moreover, the other results of [4, Section 4] are valid. In particular, the presented theory of truth conforms by [4, Theorem 4.2] to the norms presented in [5] for theories of truth.

8 8 4 Remarks Some concepts dealing with predicates and their domains are revised from those used in [4] so that they agree with the corresponding concepts in informal languages of first-order logic (cf. e.g. [1]). The assumption Let U be nonempty and consistent. of [4, Lemma 3.2] should be replaced by Assume that U is a consistent subset of D, and that U = G(U). The family of those languages which conform to classical logic is considerably larger than the family of those languages for which the theory DSTT of truth is formulated in [4]. For instance, an object language can be any language whose sentences have meanings which make them either true or false. In particular, object languages can have only a finite number of sentences. For example, let L 0 be a language formed by a sentence and its negation. If one of the sentences of L 0 is interpreted as true and the other one as false, then L 0 conforms to classical logic. But if both sentences are interpreted either as true or as false, then this interpretation contradicts with the negation rule. Thus L 0 does not conform to classical logic although it is bivalent. If a language L 0 has properties (i) (iii), it coincides with L. Thus any first-order language equipped with a consistent theory interpreted by a countable model conforms to classical logic. Mathematics, especially set theory, plays a crucial role in this note, as well as in [4]. Metaphysical necessity of pure mathematical truths is considered in [6]. References [1] Barker-Plummer, Dave, Barwise, Jon and Etchemendy, John (2011) Language, Proof and Logic, CSLI Publications, United States. [2] Chomsky, Noam (1957) Syntactic structures, The Hague: Mouton. [3] Feferman, Solomon (2012) Axiomatizing truth. Why and how? Logic, Construction, Computation (U. Berger et al. eds.) Ontos Verlag, Frankfurt, [4] Heikkilä, Seppo (2018) A mathematically derived definitional/semantical theory of truth, Nonlinear Studies, 25, 1, [5] Leitgeb, Hannes (2007) What Theories of Truth Should be Like (but Cannot be), Philosophy Compass, 2/2, [6] Leitgeb, Hannes (2018) Why Pure Mathematical Truths are Methaphysically Necessary. A Set-Theoretic Explanation, Synthese,