The Review of Symbolic Logic

Research Article

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

STEVE AWODEYa1 c1 and KOHEI KISHIDAa2 c2

a1 Carnegie Mellon University

a2 University of Pittsburgh

Abstract

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 article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

(Received June 01 2007)

Correspondence:

c1 PHILOSOPHY DEPARTMENT, CARNEGIE MELLON UNIVERSITY E-mail: awodey@cmu.edu

c2 PHILOSOPHY DEPARTMENT, UNIVERSITY OF PITTSBURGH E-mail: kok6@pitt.edu