Compositional truth with propositional tautologies and quantifier-free correctness

TL;DR

This paper investigates the conservativity of compositional truth theories with propositional tautologies and quantifier-free correctness, showing that certain assumptions lead to non-conservativity over Peano Arithmetic.

Contribution

It provides a partial answer to Cieśliński's question by analyzing the strength of truth theories with additional axioms, especially regarding quantifier-free sentences.

Findings

Assuming truth agrees with arithmetical truth on quantifier-free sentences, the theory is as strong as Δ₀-induction.

The principle of quantifier-free correctness alone is conservative.

The results clarify the boundaries of conservativity in compositional truth theories.

Abstract

Cie\'sli\'nski asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as $\Delta_0$-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.