The Finite Model Property of Quasi-transitive Modal Logic

Abstract: 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 $\mathsf{Kt}_2<sup>3$. We present a Gentzen sequent calculus $\mathsf{G}$ for $\mathsf{Kt}_2<sup>3$. The sequent calculus $\mathsf{G}$ has the finite algebra property by a finite syntactic construction. It follows that $\mathsf{Kt}_2<sup>3$ and $\mathsf{K}_2<sup>3$ have the finite model property.