Lyndon interpolation property for extensions of $\mathbf{S4}$ and intermediate propositional logics

TL;DR

This paper investigates the Lyndon interpolation property (LIP) and the uniform Lyndon interpolation property (ULIP) in extensions of S4 and intermediate propositional logics, establishing which logics possess these properties and their interrelations.

Contribution

It identifies which of the 18 known finite-height extensions of S4 have LIP and shows the equivalence of Craig, LIP, and ULIP in intermediate propositional logics.

Findings

11 out of 18 S4 extensions have LIP

7 S4 extensions do not have LIP

LIP, ULIP, and Craig interpolation are equivalent in intermediate logics

Abstract

We study the Lyndon interpolation property (LIP) and the uniform Lyndon interpolation property (ULIP) for extensions of $\mathbf{S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending $\mathbf{S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.