The McKinsey Axiom on Weakly Transitive Frames

The McKinsey axiom \((\textrm{M})\ \Box \Diamond p\rightarrow \Diamond \Box p\) $(\text{M})\square ◊p\to ◊\square p$ has a local first-order correspondent on the class of all weakly transitive frames \({{\mathcal {W}}}{{\mathcal {T}}}\) $\mathcal{W}\mathcal{T}$ . It globally corresponds to Lemmon’s condition \(({\textsf{m}}^\infty )\) $({\mathsf{m}}^{\infty })$ on \({{\mathcal {W}}}{{\mathcal {T}}}\) $\mathcal{W}\mathcal{T}$ . The formula \((\textrm{M})\) $(\text{M})$ is canonical over the weakly transitive modal logic \(\textsf{wK4}={\textsf{K}}\oplus p\wedge \Box p\rightarrow \Box \Box p\) $\mathsf{wK}\mathsf{4}=\mathsf{K}\oplus p\wedge \square p\to \square \square p$ . The modal logic \(\mathsf {wK4.1}=\textsf{wK4}\oplus \textrm{M}\) $\mathsf{wK}\mathsf{4.1}=\mathsf{wK}\mathsf{4}\oplus \text{M}$ has the finite model property. The modal logics \(\mathsf {wK4.1T}_0^n\) ${\mathsf{wK}\mathsf{4.1}\mathsf{T}}_0^n$ ( \( n>0\) $n>0$ ) form an infinite descending chain in the interval \([\mathsf {wK4.1},\mathsf {K4.1}]\) $[\mathsf{wK}\mathsf{4.1},\mathsf{K}\mathsf{4.1}]$ and each of them has the finite model property. Thus all the modal logics \(\mathsf {wK4.1}\) $\mathsf{wK}\mathsf{4.1}$ and \(\mathsf {wK4.1T}_0^n\) ${\mathsf{wK}\mathsf{4.1}\mathsf{T}}_0^n$ ( \(n>0\) $n>0$ ) are decidable.