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Spring 1997 Propositional Quantification in the Topological Semantics for S4
Philip Kremer
Notre Dame J. Formal Logic 38(2): 295-313 (Spring 1997). DOI: 10.1305/ndjfl/1039724892

Abstract

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems ${\bf S5 \pi +}$, ${\bf S4 \pi +}$, ${\bf S4.2 \pi +}$: given a Kripke frame, the quantifiers range over all the sets of possible worlds. ${\bf S5 \pi +}$ 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 ${\bf S4.2 \pi t}$, is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in ${\bf S4.2 \pi t}$. In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system ${\bf S4 \pi +}$ is recursively isomorphic to second-order logic.

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Philip Kremer. "Propositional Quantification in the Topological Semantics for S4." Notre Dame J. Formal Logic 38 (2) 295 - 313, Spring 1997. https://doi.org/10.1305/ndjfl/1039724892

Information

Published: Spring 1997
First available in Project Euclid: 12 December 2002

zbMATH: 0949.03020
MathSciNet: MR1489415
Digital Object Identifier: 10.1305/ndjfl/1039724892

Subjects:
Primary: 03B45

Rights: Copyright © 1997 University of Notre Dame

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Vol.38 • No. 2 • Spring 1997
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