The Finite Model Property of Quasi-transitive Modal Logic

The finite model property of quasi-transitive modal logic $\mathsf{K}_2^3=\mathsf{K}\oplus \Box\Box p\rightarrow \Box\Box\Box p$ is established. This modal logic is conservatively extended to the tense logic $\mathsf{Kt}_2^3$. We present a Gentzen sequent calculus $\mathsf{G}$ for $\mathsf{Kt}_2^3$. The sequent calculus $\mathsf{G}$ has the finite algebra property by a finite syntactic construction. It follows that $\mathsf{Kt}_2^3$ and $\mathsf{K}_2^3$ have the finite model property.