A proof-theoretic approach to entailment

Abstract

This paper is the technical counterpart, with a proof of the conjecture of my paper 'Entailment and Proofs' (Proc. of the Aristotelian Society, 1979). There I gave the philosophical motivation for a proof-theoretic approach to the problem of entailment. Here I preserve the spirit of that approach, with minor modifications of earlier definitions. It is hoped that the results of this paper will go some way towards a general theory of entailment and proof structure, a crucially underdeveloped aspect of extant theories of entailment, which is only gestured towards in Anderson and Belnap's encyclopaedic coverage (pp. 216-217). My own conviction is that entailment is irreducibly proof-based, and I see no need for further semantical underpinnings than those disclosed in the final Corollaries below. Before plunging into the proof theoretic workings below, the reader may benefit from some details of informal motivation. My basic aim is to avoid the Lewis paradoxes: A, ~ --B, A B v B, while retaining as much (that is, as many of the proofs) of classical logic as possible. In particular, disjunctive syllogism: A vB, ~-A B is to be retained; also, as far as possible, the transitivity of proof so vital to the development of mathematics. My basic method is to attend to "good" and "bad" features of proof in natural deduction in order to isolate some notion of Proof as a satisfactory explication of the notion of entailment. Crudely put, the Lewis paradoxes must not be Provable; but there must be enough Proofs to do all of first order mathematics. So consider the following four proofs of the first Lewis paradox in the (~, &)-fragment of classical natural deduction: