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, each of finite length, and constructed out of a countable set of symbols. A theory of syntax is assumed to introduce symbols, and rules to construct well-formed sentences. A language is called fully interpreted, if meanings are attached to its sentences so that they all are either true or false.
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⌉), where T(⌈A⌉) stands for 'the sentence (denoted by) A is true') is true for every sentence A of Ľ. The main tools to prove this result are ordinary and transfinite induction recursion methods and ZF set theory. (cf. and ).
- S. Heikkilä, A mathematically derived theory of truth and its properties. Nonlinear Studies, 25, 1, 173--189, 2018
- S. Heikkilä, A consistent theory of truth for languages which conform to classical logic. Nonlinear Studies (to appear)