The Algebras of Lewis's Counterfactuals

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.

💡 Research Summary

The paper provides a systematic logico‑algebraic treatment of David Lewis’s hierarchy of counterfactual conditional logics, a topic that has been largely neglected in the algebraic logic literature. The authors begin by distinguishing two consequence relations on Lewisian sphere models: a global (or “strong”) consequence, where inference rules may be applied to any formula, and a local (or “weak”) consequence, where the rules are restricted to theorems. Correspondingly they define two Hilbert‑style systems, GV (global) and LV (local), both sharing the same axioms (L1‑L4, classical axioms, and modus ponens) but differing in the treatment of the crucial rule (C) that propagates ordinary implication through the counterfactual connective.

The paper then develops an algebraic semantics based on Boolean algebras expanded with a binary operator ⊳ intended to model the counterfactual implication. Using the Blok‑Pigozzi framework, the authors show that GV is strongly algebraizable: the defining equations τ(x) = {x≈x⊳x} and the equivalence formulas Δ(x,y) = {x⊳y, y⊳x} translate GV’s theorems into the equational theory of the variety of “Lewis algebras,” and vice‑versa. Consequently, every GV‑derivation corresponds to a derivation in the equational logic of these algebras, and the algebraic variety provides an equivalent algebraic semantics for GV.

In contrast, LV cannot be algebraized in the Blok‑Pigozzi sense. Nevertheless, LV is shown to be the degree‑preserving logic of the same class of Lewis algebras: LV’s consequence relation preserves the truth‑degree (values in the interval