# Truth predicate

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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 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*⌉) is true for every sentence *A* of *Ľ*, where *T*(⌈*A*⌉) stands for "the sentence (denoted by) *A* is true". The main tools to prove this result are ordinary and transfinite induction, recursion methods, and ZF set theory (cf.^{[1]}
and ^{[2]}).

## See alsoEdit

## ReferencesEdit

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