# Coherence in Modal Logic

## TL;DR

This paper investigates the property of coherence in modal logic varieties, providing criteria to determine its failure, which impacts the uniform deductive interpolation property in these systems.

## Contribution

It introduces a more general criterion for coherence failure and applies it to various modal logics, expanding understanding of their algebraic properties.

## Key findings

- Coherence fails in several modal logics including K, KT, K4, and S4.
- The new criterion broadens the scope of coherence analysis in algebraic varieties.
- Failure of coherence implies failure of uniform deductive interpolation in these logics.

## Abstract

A variety is said to be coherent if the finitely generated subalgebras of its finitely presented members are also finitely presented. In a recent paper by the authors it was shown that coherence forms a key ingredient of the uniform deductive interpolation property for equational consequence in a variety, and a general criterion was given for the failure of coherence (and hence uniform deductive interpolation) in varieties of algebras with a term-definable semilattice reduct. In this paper, a more general criterion is obtained and used to prove the failure of coherence and uniform deductive interpolation for a broad family of modal logics, including K, KT, K4, and S4.