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In formal theories of truth, a truth predicate is a fundamental concept based on the sentences of a formal language as interpreted logically. That is, it formalizes the concept that is normally expressed by saying that a sentence, statement or idea "is true."
Languages which allow a truth predicateEdit
Based on 'Chomsky Definition' a language is assumed to be a countably infinite set of well-formed sentences of finite length, formed by finite number of symbols. A theory of syntax is assumed to introduce symbols, and rules for the construction of sentences. Assume that symbols contain letters, parentheses, commas, dots, constants containing natural numbers, terms containing numerals, and logical symbols including = (equal) with standard properties, ¬ (not), ∨ (or), ∧ (and),↦ (implies), ↔ (if and only if), ∀ (for all) and ∃ (exists). If P(x) is a formula with a free variable x, then ∃ x P(x) and ∀ x P(x) may be sentences.
A language is called fully interpreted, if meanings are attached to its sentences so that they all are either true or false, and the following rules are valid when A and B denote its sentences ('iff' stands for 'if and only if'): 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 if 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 false or both false, and false iff A is true and B is false or A is false and B is true. If P(x) is a formula and X is a set of terms, then P is called a predicate with domain X if P(x) is a sentence for each assignment of a term of X into x (shortly, for each x ∈ X), and if ∃ x P(x) and ∀ x P(x) exist and satisfy the following rules: ∀ x P(x) is true iff P(x) is true for every x ∈ X, and false iff P(x) is false for some x ∈ X; ∃ x P(x) is true iff P(x) is true for some x ∈ X, and false iff P(x) is false for all x ∈ X.
Any countable first-order formal language equipped with a consistent theory interpreted by a countable model, and containing natural numbers and numerals, is fully interpreted in the above sense. A classical example is the language of arithmetic with its standard model and interpretation. Another example is the first order language of set theory, the interpretation being determined by the minimal model constructed in for ZF set theory.
A fully interpreted language L which does not have a truth predicate can be extended to a fully interpreted language Ľ that contains a truth predicate T, i.e., the sentence A ↔ T(⌈A⌉) (T(⌈A⌉) stands for 'A is true') is true for every sentence A of Ľ. The main tools are ordinary and transfinite recursion methods and ZF set theory. (cf. ).
- Paul Cohen, A minimal model for set theory, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 69, pp. 537--540, 1963.
- S. Heikkilä, A mathematically derived theory of truth and its properties. DOI:10.13140/R62.2.11477.93923, 2017.
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