Relevant logics and their semantics remain viable and undamaged by Lewis's equivocation charge

Abstract

Relevant logics and their semantics are growing in notoriety, thanks (for a bad press is better than none) in large measure to the criticisms directed against them by the OxfordPrinceton axis. From Oxford criticisms have come from Scott and Copeland (both now departed from Oxford). Scott, for example, asserts, though without any of the requisite supporting argument, that "the recent semantical interpretations [for relevant logics] have as yet no adequate philosophical justification" (1974, p. 154). Scott's earlier contribution to a symposium on entailment (in 1971) strongly suggests, given his misplaced enthusiasm for Quine's criticism of modal logic, that he would endorse Quine's critique of the whole formalisation of entailment enterprise as involving a use-mention confusion in trying to treat the arrow of entailment as a systemic connective. Yet Scott had glimpsed at that stage that there was a way of engendering some illusion of understanding entailment, through a "theory of consequence" analogous to that he sketches for strict implicational system $4. The serious limitations of Scott's approach have been explained (in RLR, Postscript *), and Quine's critique of such enterprises as the formalisation of entailment have been rebutted (in Anderson and Belnap, Appendix and RLR, Chapter 1) 1. While Scott's remarks are tossed out 2 as a small sideline of a different logical industry, the main product of Copeland's shop seems to be virulent attacks on relevant logics. Yet some of the attacks of Copeland that have engaged us to the effect that relevant logic semantics do not meet the requirements (what exactly they are he fails to specify) for "an illuminating and philosophically significant semantics" no doubt give some of the substance of Scott's blanket charge as to "no adequate philosophical justification". To Copeland also we have previously replied (in PB and FR), but he has more in the publication pipeline which we have not seen (including a response to PB). From Princeton criticism has come from Lewis and from Burgess, and again more is in the pipeline. In most of what follows we focus on the case against relevant logics that Lewis has pedalled around the world, including many places in Australia, a case now published in Lewis (1982). Others have responded to Burgess3; but the dialectic that has begun, especially concerning Material Detachment, is unlikely given the thorniness of the problems to come to an early end. Lewis's effort may look, like Scott's, like a sideline. It has that same "Look fellows, no hands" air about it, though similarly a gossamer. But the matter is serious: for if the semantical picture of relevant logic stands up, Lewis's philosophical position goes down the tube. One important reason is, in brief, that there is no plausible way of representing the inconsistent and radically incomplete worlds of relevant semantics among the complete possible worlds that Lewis countenances, of viewing them as "concrete worlds" just, or much, like our own except that we're not actually in them, and "differing not in kind but only in what goes on in them". That is why Lewis has resort instead to (an adaption of) Jaskowski's method, fragmentation as it is now pleasantly called, which, to the limited extent that it works, enables inconsistent worlds to be avoided, inconsistent worlds fragmenting into sets of overlapping possible worlds. But the appealing idea of partitioning an inconsistent theory into maximal consistent subtheories, encounters really serious impediments, especially once the finite case (investigated to some extent by Jennings and Schotch) is left behind. For one thing there is no effective procedure for accomplishing this in general. For another the consistent subtheories that would result would be subject to most unwelcome limitations, regarding for instance what we can satisfactorily express. Do not get the impression that we dislike consistent theories. While we have suggested, in past work (e.g. RLR, Section 1.6), that inconsistency might be ineluctable under some circumstances indeed, that it might be ineluctable in a true understanding of the world we are very far from preferring inconsistent set-ups when consistent ones will do. Granted, we are not without sympathy for Walt Whitman: "Do I contradict myself? Very well then I contradict myself. [I am large, I contain multitudes] ". But we are more than content to chop the contradictions out of these