  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.
