Saturation properties for compositional truth with propositional correctness

TL;DR

This paper investigates the impact of propositional soundness on compositional truth theories, revealing that it enforces saturation-like properties that limit their conservativity over base arithmetical theories.

Contribution

It demonstrates that propositional soundness introduces saturation properties, providing new insights into the limitations of conservativity in compositional truth theories.

Findings

Propositional soundness induces saturation-like properties.

These properties impose limitations on conservativity.

The results clarify the relationship between truth principles and model properties.

Abstract

It is an open question whether compositional truth with the principle of propositional soundness ,,all arithmetical sentences which are propositional tautologies are true'' is conservative over its arithmetical base theory. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the underlying model, thus showing significant limitations to the possible conservativity proof.