Topological Completeness of First-Order Modal Logics

As McKinsey and Tarski [20] 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. This topological interpretation was recently extended in a natural way to arbitrary theories of full first-order logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full first-order S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems.