Propositional Quantification in the Topological Semantics for S4

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.