A Basic Formal Equational Predicate Logic

We present the details of a formalization of Equational Predicate Logic based on a propositional version of the Leibniz rule (PSL—Propositional Strong Leibniz), EQN (equanimity) and TR (transitivity). All the above rules are “strong”, that is, they are applicable to arbitrary premises (not just to absolute theorems). It is shown that a strong no-capture Leibniz (SLCS—Strong Leibniz with “Contextual Substitution”), and a weak full-capture version (Weak-Leibniz with Uniform Substitution, or WLUS) are derived rules. “Weak” means that the rule is only applicable on absolutely deducible premises. We also derive general rules MON (monotonicity) and AMON (antimonotonicity), which allow us to “calculate” appropriate conclusions ⊢ C[p\A] ⇒ C[p\B] or ⊢ C[p\A] ⇐ C[p\B] from the assumption ⊢ A ⇒ B. Introduction. This note builds further on [To], where the logical “calculus” of Equational (Predicate) Logic outlined in [GS1] was formalized and shown to be sound and complete. We propose here a simpler formalization than the one in [To], basing the proof-apparatus solely on propositional rules of inference—one of which, of course, is a version of “Leibniz”. This entails an unconstrained Deduction Theorem (contrast with [To]), which in turn further simplifies the steps of our reasoning. While our “foundations” include “just” a propositional version of Leibniz, we show that there are derived rules valid in the logic, which allow the use of Leibniz-style substitution within the scope of a quantifier. We also address one “weakness”—to which David Gries has already called attention in [Gr]—of the current literature ([DSc, GS1]) on equational or calculational reasoning. That is, while it is customary to mix =-steps (that is, an application of a conjunctional ≡) and ⇒-steps (that is, an application of a conjunctional ⇒) in a calculational proof, and while we have well documented rules to handle the former, yet the latter type of step normally seems to rely on a compendium of ad hoc rules. We hope to have contributed towards remedying this state of affairs, as we present a unifying, yet simple and rigorous way to understand, ascertain validity, and therefore annotate and utilize ⇒-steps, using the rules monotonicity and antimonotonicity. We conclude with a section on soundness and completeness of the proposed Logic. The term “basic” in the title is meant to convey that we include no more than what is necessary to lay the foundations. In particular, examples that illustrate the power of calculational reasoning were left out.