David Lewis employed \({\Box }{\rightarrow }\) and \(\preccurlyeq \) as the primitive connectives, respectively, to establish two different kinds of conditional logic systems, which can be demonstrated to be equivalent. Unfortunately, Lewis and his successors relied solely on an intuitive sphere semantic model to ascertain their equivalence, failing to provide a formal proof. A formal clarification of the relationship between these different logical systems is pivotal for intuitively grasping and comprehending them and their interconnections. Hence, the aim of this paper is to provide a rigorous equivalence proof between Lewis’ two kinds of conditional systems: CO and C1, as well as V and VC. Through our proof, we show that the Connex axiom is redundant within System V and give its derivation from the other rules and axioms. This indicates that the connective \(\preccurlyeq \) is more foundational than Lewis anticipated, and has precedence over other conditional connectives. Furthermore, we propose four semantics for \(\preccurlyeq \) , some of which were not previously posited by Lewis, and examine the potential for extending \(\preccurlyeq \) to other logics.
