In formal theories of

truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...

, a truth predicate is a fundamental concept based on the

sentences The ''Sentences'' (. ) is a compendium of Christian theology written by Peter Lombard around 1150. It was the most important religious textbook of the Middle Ages. Background The sentence genre emerged from works like Prosper of Aquitaine's ...

of a

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

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 Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for a ...

, recursion methods, and ZF set theory (cf. and S. Heikkilä, A consistent theory of truth for languages which conform to classical logic. Nonlinear Studies (to appear)).

References

{{Ling-stub Mathematical logic Theories of truth Predicate

Languages which allow a truth predicate

See also

References