Notre Dame Journal of Formal Logic

Propositional Quantification in the Topological Semantics for S4

Philip Kremer

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 38, Number 2 (1997), 295-313.

Dates
First available in Project Euclid: 12 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1039724892

Digital Object Identifier
doi:10.1305/ndjfl/1039724892

Mathematical Reviews number (MathSciNet)
MR1489415

Zentralblatt MATH identifier
0949.03020

Subjects
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}

Citation

Kremer, Philip. Propositional Quantification in the Topological Semantics for S4. Notre Dame J. Formal Logic 38 (1997), no. 2, 295--313. doi:10.1305/ndjfl/1039724892. https://projecteuclid.org/euclid.ndjfl/1039724892


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