Models of Positive Truth

TL;DR

This paper strengthens the understanding of the semantic strength of theories of truth, showing that adding internal induction axioms increases their strength and providing an axiomatic theory aligning with philosophical criteria.

Contribution

It demonstrates the non-conservativity of PT^- with internal induction and introduces an axiomatic truth theory satisfying philosophical requirements.

Findings

PT^- + INT(tot) is semantically weaker than PT^- with full internal induction

The latter is not relatively truth definable in the former

An axiomatic truth theory meeting Fischer and Horsten's criteria is provided

Abstract

This paper is a follow-up to "Models of PT${}^-$ with internal induction for total formulae." We give a strenghtening of the main result on the semantical non-conservativity of the theory of PT${}^-$ with internal induction for total formulae (PT${}^- +$ INT(tot)). We show that if to PT${}^-$ the axiom of internal induction for all arithmetical formulae is added (PT${}^-$), then this theory is semantically stronger than PT${}^- +$ INT(tot). In particular the latter is not relatively truth definable (in the sense of Fujimoto) in the former. Last but not least we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in "The expressive power of truth."