# Truth predicate

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 predicate

Based on 'Chomsky Definition' a language is assumed to be a countable set of sentences of finite length, formed by symbols containing letters, parentheses, logical symbols ¬ (not), ∨ (or), ∧ (and),↦ (implies), ↔ (if and only if), ∀ (for all) and ∃ (exists), = (equal) with its standard properties, all natural numbers and terms, including numerals. If P is a predicate of a language, and a set X of terms is a domain of P, then P(x) is a sentence of that language for each x ∈ X (more precisely, for each assignment of a term of X into x)

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 is a predicate and X denotes its domain, then ∀ 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 [1] 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. Main tools are ordinary and transfinite recursion methods and ZF set theory. (cf. [2]).

[1] Paul Cohen, A minimal model for set theory, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 69, pp. 537--540, 1963.

[2] S. Heikkilä, A mathematically derived theory of truth and its properties. DOI:10.13140/R62.2.11477.93923, 2017.