## Notre Dame Journal of Formal Logic

### A Note on Algebraic Semantics for $\mathsf{S5}$ with Propositional Quantifiers

Wesley H. Holliday

#### 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 $\mathsf{L}$?

#### Article information

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

Dates
Accepted: 19 January 2017
First available in Project Euclid: 9 May 2019

https://projecteuclid.org/euclid.ndjfl/1557388819

Digital Object Identifier
doi:10.1215/00294527-2019-0001

Mathematical Reviews number (MathSciNet)
MR3952235

Zentralblatt MATH identifier
07096540

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1557388819

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