## Completeness of second-order propositional S4 and H in topological semantics

We add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionsitic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

Keywords: Second-order propositional logic, modal logic, intuitionistic logic, topological semantics.