TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this paper the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 firstorder modal logic is complete with respect to such topological semantics. It has been known since the work of McKinsey and Tarski [?] that, by extending the Stone representation theorem for Boolean algebras, topological spaces provide semantics to propositional modal logic. Specifically, a necessity operator obeying the rules of the system S4 can be interpreted by the interior operation in a topological space. This result, however, is limited to propositional modal logic. The aim of this paper is to show how the topological interpretation can be extended in a very natural way to first-order modal logic. 1. Topological Semantics for Propositional Modal Logic Let us review the topological semantics for propositional S4. 1.1. The System S4 of Propositional Modal Logic. Modal logic is the study of logic in which the words “necessary” and “possible” appear in statements such as • It is necessary that the square of an integer is not negative. • It is possible that there are more than 8 planets. The history of modal logic is as old as that of the study of logic in general, and can be traced back to the time of Aristotle. The contemporary study of modal logic typically treats modal expressions as sentential operators, in the same way as ¬ is treated. That is, for each formula φ of propositional logic, the following are again A grateful acknowledgment goes to inspiring discussions with and helpful comments by Horacio Arlo-Costa, Nuel Belnap, Johan van Benthem, Mark Hinchliff, Paul Hovda, Ken Manders, Eric Pacuit, Rohit Parikh, Dana Scott, and especially Guram Bezhanishvili, Silvio Ghilardi and Rob Goldblatt as well as an anonymous referee for their accurate suggestions which improved Section ??. We also thank the organizers, Aldo Antonelli, Alasdair Urquhart, and Richard Zach, of the Banff Workshop “Mathematical Methods in Philosophy” for the opportunity to present this research. Philosophy Department, Carnegie Mellon University; awodey@cmu.edu. Philosophy Department, University of Pittsburgh; kok6@pitt.edu.