Theories of truth for countable languages which conform to classical logic

TL;DR

This paper demonstrates that any countable classical logic language can be extended to include a definitional and semantical truth theory, integrating truth and non-truth predicates with equivalence under meaning-based valuation.

Contribution

It introduces a method to extend countable classical languages with comprehensive truth theories that include both truth and non-truth predicates, ensuring semantic valuation.

Findings

Existence of extensions with definitional truth theories

Extensions can have a semantic truth theory based on meaning valuation

Truth and non-truth predicates coexist in the theories

Abstract

Every countable language which conforms to classical logic is shown to have an extension which conforms to classical logic, and has a definitional theory of truth. That extension has a semantical theory of truth, if every sentence of the object language is valuated by its meaning either as true or as false. These theories contain both a truth predicate and a non-truth predicate. Theories are equivalent when the sentences of the object language are valuated by their meanings.