Fischer-Servi logic does not have interpolation

TL;DR

This paper proves that Fischer-Servi logic and some of its extensions lack the Craig interpolation property by demonstrating the absence of the amalgamation property in related algebraic structures.

Contribution

It establishes the non-existence of interpolation in Fischer-Servi logic and extends this result to several of its logical extensions.

Findings

Fischer-Servi logic $\\mathsf{IK}$ lacks Craig interpolation.

The associated class of modal Heyting algebras does not have the amalgamation property.

Extensions like $\\mathsf{IT}$, $\\mathsf{IK4}$, $\\mathsf{IS4}$, and $\\mathsf{IGL}$ also lack interpolation.

Abstract

We prove that the Fischer-Servi logic $\mathsf{IK}$ does not have the (Craig) interpolation property. This is obtained by showing that the corresponding class of modal Heyting algebras lacks the amalgamation property. We also generalize this result to some extensions of the Fischer-Servi logic such as $\mathsf{IT}$, $\mathsf{IK4}$, $\mathsf{IS4}$, and $\mathsf{IGL}$.