On the Soundness and Completeness of Equational Predicate Logics

This paper is about the elementary metatheory of equational predicate logic—that is, first‐order logic that uses Leibniz's substitution of ‘equals for equals’ as a primary rule of inference. We present here two different ‘faithful’ formalizations of this logic and prove that both versions are sound and complete. A by‐product of this study is a proof that the ‘full’ Leibniz rule is strictly stronger than the ‘no‐capture’ Leibniz rule. We obtain this result here for a complete Logic. We also show that under some reasonable conditions, propositional Leibniz, no‐capture Leibniz, and a full‐capture version are all equivalent, provided that the latter is restricted to act on universally valid premises whenever capture is allowed.