The persistence principle over weak interpretability logic

TL;DR

This paper investigates the properties of a logic formed by adding the persistence principle to weak interpretability logic, establishing key features like fixed points, interpolation, and semantics.

Contribution

It introduces a new logic with the persistence principle, proves its fixed point property, cut-elimination, interpolation, and provides soundness and completeness results.

Findings

Logic has a weak fixed point property.

Proved cut-elimination and Craig interpolation.

Established finite frame and arithmetical completeness.

Abstract

We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut-elimination theorem for it. As a consequence, we prove that the logic enjoys the Craig interpolation property. Thirdly, we show that the logic is the natural basis of a generalization of simplified Veltman semantics, and prove that it has the finite frame property with respect to that semantics. Finally, we prove that it is sound and complete with respect to some appropriate arithmetical semantics.