Finite axiomatization of $\textbf{GL}\times\textbf{S5}$ and $\textbf{Grz}\times\textbf{S5}$

Abstract

We prove that $\mathbf{GL} \times \mathbf{S5}$ is product matching, and that $\mathbf{Grz} \times \mathbf{S5}$ is axiomatizable by adding to $[\mathbf{Grz},\mathbf{S5}]$ the G\"odel translation of the monadic Casari formula. This settles the question of the finite axiomatizability of these logics posed by Gabbay and Shehtman (1998).