On Axiomatization of Lewis' Conditional Logics

TL;DR

This paper clarifies the differences between Nute's and Lewis's axiomatic systems for conditional logics, proposes new axiomatizations avoiding certain rules, and resolves a longstanding puzzle about disjunctive antecedents.

Contribution

It introduces new axiomatizations for Lewis' conditional logics that do not rely on CSO, RCEA, or interchange rules, and addresses a controversial axiom in the field.

Findings

Nute's systems are not equivalent to Lewis' original systems.

New axioms based on cautious monotonicity and cautious cut are proposed.

A solution to the disjunctive antecedent simplification puzzle is provided.

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.