The McKinsey Axiom on Weakly Transitive Frames

The McKinsey axiom \((\textrm{M})\ \Box \Diamond p\rightarrow \Diamond \Box p\) has a local first-order correspondent on the class of all weakly transitive frames \({{\mathcal {W}}}{{\mathcal {T}}}\). It globally corresponds to Lemmon’s condition \(({\textsf{m}}^\infty )\) on \({{\mathcal {W}}}{{\mathcal {T}}}\). The formula \((\textrm{M})\) is canonical over the weakly transitive modal logic \(\textsf{wK4}={\textsf{K}}\oplus p\wedge \Box p\rightarrow \Box \Box p\). The modal logic \(\mathsf {wK4.1}=\textsf{wK4}\oplus \textrm{M}\) 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.