The McKinsey Axiom on Weakly Transitive Frames

Authors: Qian Chen and Minghui Ma
Published in: Studia Logica (2024). DOI: https://doi.org/10.1007/s11225-024-10145-x

Abstract

The McKinsey axiom $(\mathrm{M})\ \Box\Diamond p\to \Diamond\Box p$ has a local first-order correspondent on the class of all weakly transitive frames $\mathsf{WT}$. It globally corresponds to Lemmon’s condition $(\mathsf{m}^\infty)$ on $\mathsf{WT}$. The formula $(\mathrm{M})$ is canonical over the weakly transitive modal logic $\mathsf{wK4}=\mathsf{K}\oplus p\wedge\Box p\to \Box\Box p$. The modal logic $\mathsf{wK4.1}=\mathsf{wK4}\oplus \mathrm{M}$ has the finite model property. The modal logics $\mathsf{wK4.1T}_0^n$ ($ n>0$) form an infinite descending chain in the interval $[\mathsf{wK4.1},\mathsf{K4.1}]$ and each of them has the finite model property. Thus all the modal logics $\mathsf{wK4.1}$ and $\mathsf{wK4.1T}_0^n$ ($n>0$) are decidable.