Propositional Quantification in the Topological Semantics for S4

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems S5π+, S4π+, S4.2π+: given a Kripke frame, the quantifiers range over all the sets of possible worlds. S5π+ 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 S4πt, is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in S4πt. In the course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to second-order logic.