Notre Dame Journal of Formal Logic 38 (2):295-313 (1997)

Philip Kremer
University of Toronto at Scarborough
Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in . In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system is recursively isomorphic to second-order logic
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DOI 10.1305/ndjfl/1039724892
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References found in this work BETA

A Completeness Theorem in Modal Logic.Saul A. Kripke - 1959 - Journal of Symbolic Logic 24 (1):1-14.
Semantical Analysis of Modal Logic I. Normal Propositional Calculi.Saul A. Kripke - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (5‐6):67-96.
Completeness in the Theory of Types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
Propositional Quantifiers in Modal Logic.Kit Fine - 1970 - Theoria 36 (3):336-346.

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Citations of this work BETA

A Note on Algebraic Semantics for S5 with Propositional Quantifiers.Wesley H. Holliday - 2019 - Notre Dame Journal of Formal Logic 60 (2):311-332.
Pitts' Quantifiers Are Not Topological Quantification.Tomasz Połacik - 1998 - Notre Dame Journal of Formal Logic 39 (4):531-544.

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