A calculus for modal compact Hausdorff spaces

The symmetric strict implication calculus S(2)ICis a modal calculus for compact Hausdorff spaces. This is established throughde Vries duality, linking compact Hausdorff spaces with de Vries algebras-complete Boolean algebras equipped with aspecial relation. Modal compact Hausdorff spaces are compact Hausdorff spaces enriched with a continuous relation. Thesespaces correspond, via modalized de Vries duality, to upper continuous modal de Vries algebras. In this paper, we introducethe modal symmetric strict implication calculus (MSIC)-I-2, which extends (SIC)-I-2. We prove that MS(2)ICis strongly sound andcomplete with respect to upper continuous modal de Vries algebras, thereby providing a logical calculus for modal compactHausdorff spaces. We also develop a relational semantics for MS(2)ICthat we employ to show admissibility of various Pi(2)-rules in this system