Notre Dame Journal of Formal Logic

A Note on Algebraic Semantics for S5 with Propositional Quantifiers

Wesley H. Holliday

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In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5Π extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5Π 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 L can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for L?

Article information

Notre Dame J. Formal Logic, Volume 60, Number 2 (2019), 311-332.

Received: 22 December 2016
Accepted: 19 January 2017
First available in Project Euclid: 9 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Secondary: 03C80: Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48] 03G05: Boolean algebras [See also 06Exx]

modal logic propositional quantifiers algebraic semantics monadic algebras MacNeille completion


Holliday, Wesley H. A Note on Algebraic Semantics for $\mathsf{S5}$ with Propositional Quantifiers. Notre Dame J. Formal Logic 60 (2019), no. 2, 311--332. doi:10.1215/00294527-2019-0001.

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