A Note on Algebraic Semantics for S5 with Propositional Quantifiers

Abstract

In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5$\Pi$ extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5$\Pi$ with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: For which normal modal logics $\mathsf{L}$ can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for $\mathsf{L}$?