Knocking Down Boxes: The FMP for $\mathbf{K}\oplus {\square }^{m+k}p\to {\square }^{m}p$

Abstract

It is a long-standing open problem whether modal logics of the form $\mathbf{K}\oplus {\square }^{n}p\to {\square }^{m}p$ for $n>m>1$ have the finite model property (FMP). We solve this by showing that any modal logic axiomatized by formulas of the form ${\square }^{n}p\to {\square }^{m}p$ where $n>m>1$ has the FMP.

11 pages