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.

Abstract A general method is provided whereby bizarre revisions of consistent theories with respect to contingent sentences that they refute can be delivered by revision-functions satisfying both the basic and the supplementary postulates of the AGM-theory of theory-revision.

Journal Article Editorial Get access NEIL TENNANT NEIL TENNANT Executive Editor Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 1, Issue 4, September 1991, Pages 427–430, https://doi.org/10.1093/logcom/1.4.427 Published: 01 September 1991

Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A. Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism). But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.