THE ALGEBRAS OF LEWIS’S COUNTERFACTUALS: DUALITY THEORY

Abstract This paper explores the mathematical connections between the algebraic and relational semantics of Lewis’s logics for counterfactual conditionals. Specifically, we introduce topological variants of Lewis’s well-known possible-worlds semantics—based on spheres, selection functions, and orders—and establish duality results with respect to varieties of Boolean algebras equipped with a counterfactual operator, which serve as the equivalent algebraic semantics of Lewis’s main systems. These results aim to provide a solid mathematical foundation for the study of Lewis’s logics, and offer a new perspective on the most well-known possible worlds-based models. In particular, we write explicit proofs for several results that are often assumed without proof in the literature. Leveraging these duality results, we also derive alternative proofs of strong completeness for Lewis’s variably strict conditional logics with respect to their intended models, and clarify the role of the limit assumption in sphere semantics.