Natural deduction and sequent calculus for intuitionistic relevant logic

Abstract

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.