Bimodal logics with a `weakly connected' component without the finite model property

TL;DR

This paper investigates the finite model property in bimodal logics with a weakly connected component, showing that such logics often lack the fmp, especially when components are universally axiomatisable or have frames of limited depth.

Contribution

It demonstrates that commutators with a weakly connected component frequently do not possess the finite model property, extending previous results and covering cases with limited frame depth.

Findings

Commutators with weakly connected components often lack the fmp.

The positive results for finitely axiomatisable components do not extend to universally axiomatisable logics.

Even partial commutativity can lead to infinite frames in certain bimodal logics.

Abstract

There are two known general results on the finite model property (fmp) of commutators [L,L'] (bimodal logics with commuting and confluent modalities). If L is finitely axiomatisable by modal formulas having universal Horn first-order correspondents, then both [L,K] and [L,S5] are determined by classes of frames that admit filtration, and so have the fmp. On the negative side, if both L and L' are determined by transitive frames and have frames of arbitrarily large depth, then [L,L'] does not have the fmp. In this paper we show that commutators with a `weakly connected' component often lack the fmp. Our results imply that the above positive result does not generalise to universally axiomatisable component logics, and even commutators without `transitive' components such as [K.3,K] can lack the fmp. We also generalise the above negative result to cases where one of the component logics has frames of depth one only, such as [S4.3,S5] and the decidable product logic S4.3xS5. We also show cases when already half of commutativity is enough to force infinite frames.