## 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

#### 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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