A strict implication calculus for compact Hausdorff spaces

We introduce a simple modal calculus for compact Hausdorff spaces. The language of our system extends that of propositional logic with a strict implication connective, which, as shown in earlier work, algebraically corresponds to the notion of a subordination on Boolean algebras. Our base system is a strict implication calculus SIC, to which we associate a variety SIA of strict implication algebras. We also study the symmetric strict implication calculus S2IC, which is an extension of SIC, and prove that S²IC is strongly sound and complete with respect to de Vries algebras. By de Vries duality, this yields completeness of S²IC with respect to compact Hausdorff spaces. Since some of the defining axioms of de Vries algebras are Π₂-sentences, we develop the corresponding theory of non-standard rules, which we term Π₂-rules. We study the resulting inductive elementary classes of algebras, and give a general criterion of admissibility for Π₂-rules. We also compare our approach to approaches in the literature that are related to our work.