A mathematical theory of truth and an application to the regress problem

TL;DR

This paper introduces a formal mathematical framework for languages with truth predicates, addressing the regress problem and aligning with philosophical norms for theories of truth.

Contribution

It develops a mathematical theory of truth for a class of formal languages, avoiding infinite regress and extending prior philosophical frameworks.

Findings

MTT creates languages with their own truth predicates

MTT conforms to established norms for theories of truth

Provides a framework to study the regress problem

Abstract

In this paper a class of languages which are formal enough for mathematical reasoning is introduced. First-order formal languages containing natural numbers and numerals belong to that class. Its languages are called mathematically agreeable (shortly MA). Languages containing a given MA language L, and being sublanguages of L augmended by a monadic predicate are constructed. A mathematical theory of truth (shortly MTT) is formulated for some of these languages. MTT makes them MA languages which posses their own truth predicates. MTT is shown to conform well with the eight norms presented for theories of truth in 'What Theories of Truth Should be Like (but Cannot be)', by Hannes Leitgeb. MTT is free from infinite regress, providing a proper framework to study the regress problem. Main tools used in proofs are Zermelo-Fraenkel (ZF) set theory and classical logic.