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Propositional Quantification in Bimodal \(\mathbf {S5}\)

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Abstract

Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions.

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Acknowledgements

Thanks to Jeremy Goodman for prompting me to think about the complexity of \(\Lambda _{\mathsf {PE}}\), and to Timothy Williamson and three anonymous referees for comments on drafts of this paper.

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  1. University of Oslo, Postboks 1020 Blindern, 0315, Oslo, Norway

    Peter Fritz

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  1. Peter Fritz
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Correspondence to Peter Fritz.

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Fritz, P. Propositional Quantification in Bimodal \(\mathbf {S5}\). Erkenn 85, 455–465 (2020). https://doi.org/10.1007/s10670-018-0035-3

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  • DOI: https://doi.org/10.1007/s10670-018-0035-3

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