In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5 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 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 can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for ?
"A Note on Algebraic Semantics for with Propositional Quantifiers." Notre Dame J. Formal Logic 60 (2) 311 - 332, May 2019. https://doi.org/10.1215/00294527-2019-0001