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On the Soundness and Completeness of Equational Predicate Logics

We present two different formalizations 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 prove that both versions are sound and complete. A by-product of this study is an alternative proof to that contained in [GS3], that the “full” Leibniz rule is strictly stronger than the “no-capture” Leibniz rule, this result obtained 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.