A calculus for modal compact Hausdorff spaces

Abstract The symmetric strict implication calculus $\mathsf{S}^{2}\mathsf{IC}$ is a modal calculus for compact Hausdorff spaces. This is established through de Vries duality, linking compact Hausdorff spaces with de Vries algebras—complete Boolean algebras equipped with a special relation. Modal compact Hausdorff spaces are compact Hausdorff spaces enriched with a continuous relation. These spaces correspond, via modalized de Vries duality, to upper continuous modal de Vries algebras. In this paper, we introduce the modal symmetric strict implication calculus $\mathsf{MS}^{2}\mathsf{IC}$, which extends $\mathsf{S}^{2}\mathsf{IC}$. We prove that $\mathsf{MS}^{2}\mathsf{IC}$ is strongly sound and complete with respect to upper continuous modal de Vries algebras, thereby providing a logical calculus for modal compact Hausdorff spaces. We also develop a relational semantics for $\mathsf{MS}^{2}\mathsf{IC}$ that we employ to show admissibility of various $\Pi_{2}$-rules in this system.