The Algebras of Lewis's Counterfactuals

TL;DR

This paper provides a comprehensive algebraic analysis of Lewis's counterfactual logics, introducing new axiomatizations, exploring their algebraizability, and establishing dualities with topological models to deepen understanding of their mathematical foundations.

Contribution

It introduces novel finite axiomatizations for Lewis's logics, analyzes their algebraizability, and establishes dualities with topological spaces, advancing the mathematical understanding of counterfactual logics.

Findings

Global consequence relation is strongly algebraizable with Boolean algebras.

Local consequence relation is not algebraizable but characterized as degree-preserving logic.

Established dualities with topological models, including sphere models and Stone spaces.

Abstract

The logico-algebraic study of Lewis's hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work aims to fill this gap by providing a comprehensive logico-algebraic analysis of Lewis's logics. We begin by introducing novel finite axiomatizations for varying strengths of Lewis's logics, distinguishing between global and local consequence relations on Lewisian sphere models. We then demonstrate that the global consequence relation is strongly algebraizable in terms of a specific class of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local consequence relation is generally not algebraizable, although it can be characterized as the degree-preserving logic over the same algebraic models. Further, we delve into the algebraic semantics of Lewis's logics, developing two dual equivalences with respect to particular topological spaces. In more details, we show a duality with respect to the topological version of Lewis's sphere models, and also with respect to Stone spaces with a selection function; using the latter, we demonstrate the strong completeness of Lewis's logics with respect to sphere models. Finally, we draw some considerations concerning the limit assumption over sphere models.