Perfect proofs at first order

Abstract

In this note we extend a remarkable result of Brauer (2024, Journal of Logic and Computation) concerning propositional Classical Core Logic. We show that it holds also at first order. This affords a soundness and completeness result for Classical Core Logic. The $\mathbb{C}^{+}$-provable sequents are exactly those that are uniform substitution instances of perfectly valid sequents, i.e. sequents that are valid and that need every one of their sentences in order to be so. Brauer (2020, Review of Symbolic Logic, 13, 436–457) showed that the notion of perfect validity itself is unaxiomatizable. In the Appendix we use his method to show that our notion of relevant validity in Tennant (2024, Philosophia Mathematica) is likewise unaxiomatizable. It would appear that the taking of substitution instances is an essential ingredient in the construction of a semantical relation of consequence that will be axiomatizable—and indeed, by the rules of proof for Classical Core Logic.