Logic on the Australian plan
This paper describes the analysis of the negation of relevant logics in terms of the Routley *-operation, and sketches a philosophical underpinning for it. Basic to this is the idea of a dual theory. A theory is a set of sentences closed under entailment. For any theory, T, the dual theory, \(T^*\), is just that theory which asserts what T fails to deny: \(A\in T^*\) iff \(\sim A\not\in T\). The ''truth conditions'' for \(\sim\) now follow simply by contraposition: \(\sim A\in T\) iff \(A\not\in T^*\). In an ''ideal'' situation, where T is maximally consistent, \(T=T^*\), and the familiar truth conditions for negation emerge: \(\sim A\in T\) iff \(A\not\in T\). There are, however, many situations that are not ideal and in which T is non-trivially inconsistent or incomplete. The classical situation is therefore only a special case. A crucial question now is how the theories in question are to be constructed. If theories are taken to be sets of truths, then the view is committed to the possibility that truth may be inconsistent or incomplete. This view the paper categorically rejects. Instead, theories are to be taken to be our theories about something or other. But now the following problems arise: i) It is crucial for this account that there are ''real life'' theories that are inconsistent and incomplete. This is problematic. It may be denied that there are non-trivial inconsistent theories; and though it may be admitted that there are incomplete theories, it may be denied that their duals are theories. The paper points out that we often reason in situations where our data is inconsistent or incomplete; but this does not justify us in thinking of the situations as closed under entailment; for characteristically the reasoning that goes on in these situations is non-monotonic: after a certain amount of deduction we perform some back- tracking and readjust our premises. ii) The paper argues, correctly, against giving an epistemic interpretation of certain semantics of relevant logics (as has sometimes been suggested), on the ground that truth and falsity are not epistemic notions. But since theories, on this conception, are equally epistemic entities, parallel objections would seem to apply. iii) The paper does start to develop a reply to this objection: that there is no truth and falsity independent of our best (long run) theories. Whatever one is to make of this pragmatism (and of course, there are well known arguments against it), it should be noted that this view tends to undercut other parts of the paper. Once we accept that there are inconsistent/incomplete theories, we have to concede the possibility (as the paper does) that our best long run theories may be like this. Hence truth (or some truths - for we have no guarantee that the best long run theory is unique), it turns out after all, may well be inconsistent and incomplete.
scientific article; zbMATH DE number 3993532