The Classical Logic of Relevant Logicians

Abstract

Are relevant logicians committed to accepting as formally correct the same natural language arguments as classical logicians? I have elsewhere suggested that they were (in my 74). I want to argue for this suggestion after discussing some terminology. By ‘relevant logician’ I mean a logician who accepts a natural language argument A being formally correct on a classical sentential analysis if and only if a classical sentential analysis of A produces a sentential inference form (P1,P2,…,Pn ∴Q) such that ((P1 & P2 &…& Pn) ⊃ Q) is a tautological entailment in the sense of Anderson and Belnap1. We give a classical sentential analysis of A if we represent a form of A as (P1,P2,…,Pn ∴Q) where ∴ symbolises ‘therefore’ and the premiss forms Pi and the conclusion form Q are in some notation suitable for classical sentential logic in which (A ⊃ B) is (~A ⋁ B). Usually I shall regard the premiss forms conjoined and talk of inference forms (P ∴ Q). I use ‘formally correct’ because a relevant logician does not accept an argument as meeting the formal conditions for deductive adequacy simply if a classical sentential analysis of it produces a classically valid form. (In the remainder of this essay ‘valid’ will mean ‘classically valid’.) The relevant logician, of course, requires that a form F of an argument A in virtue of which A is certified as meeting the formal conditions of deductive adequacy be valid and also at least be such that in any natural language interpretation of F some of the topics talked about in the premisses will be talked about in the conclusion2. In other words, the relevant logician demands that the form in virtue of which an argument is accepted guarantee that there be relevance between the premisses and the conclusion.