On Axiomatization of Lewis' Conditional Logics

Abstract: This paper first shows that the popular axiomatic systems proposed by Nute for Lewis' conditional logics are not equivalent to Lewis' original systems. In particular, the axiom CA which is derivable in Lewis' systems is not derivable in Nute's systems. Then the paper proposes a new set of axiomatizations for Lewis' conditional logics, without using CSO, or RCEA, or the rule of interchange of logical equivalents. Instead, the new axiomatizations adopt two axioms which correspond to cautious monotonicity and cautious cut in nonmonotonic logics, respectively. Finally, the paper gives a simple resolution to a puzzle about the controversial axiom of simplification of disjunctive antecedents, using a long neglected axiom in one of Lewis' systems for conditional logics.