In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted (...) by a number of authors under the heading of abstract entailment relations where it has found an important role in both algebra and theoretical computer science. In this essay we go back to the beginnings, as presented by Scott, in order to make some comments about Scott’s cut rule, and show how much of Scott’s main result may be applied to the case of single-conclusion logic. (shrink)

. This paper examines the underpinnings of the preservationist approach to characterizing inference relations. Starting with a critique of the ‘truth-preservation’ semantic paradigm, we discuss the merits of characterizing an inference relation in terms of preserving consistency. Finally we turn our attention to the generalization of consistency introduced in the early work of Jennings and Schotch, namely the concept of level.

H. B. Smith, Professor of Philosophy at the influential 'Pennsylvania School' was (roughly) a contemporary of C. I. Lewis who was similarly interested in a proper account of 'implication'. His research also led him into the study of modal logic but in a different direction than Lewis was led. His account of modal logic does not lend itself as readily as Lewis' to the received 'possible worlds' semantics, so that the Smith approach was a casualty rather than a beneficiary of (...) the renewed interest in modality. In this essay we present some of the main points of the Smith approach, in a new guise. (shrink)

"The best known approaches to "reasoning with inconsistent data" require a logical framework which is decidedly non-classical. An alternative is presented here, beginning with some motivation which has been surprised in the work of C.I. Lewis, which does not require ripping great swatches from the fabric of classical logic. In effect, the position taken in this essay is representative of an approach in which one assumes the correctness of classical methods excepting only the cases in which the premise set is (...) inconsistent. (shrink)

The standard semantics for sentential modal logics uses a truth condition for necessity which first appeared in the early 1950s. in this paper the status of that condition is investigated and a more general condition is proposed. in addition to meeting certain natural adequacy criteria, the more general condition allows one to capture logics like s1 and s0.9 in a way which brings together the work of segerberg and cresswell.

This essay attempts to implement epistemic logic through a non-classical inference relation. Given that relation, an account of '(the individual) a knows that A' is constructed as an unfamiliar non-normal modal logic. One advantage to this approach is a new analysis of the skeptical argument.

In the fourteenth century, Duns Scotus suggested that the proper analysis of modality required not just moments of time but also “moments of nature”. In making this suggestion, he broke with an influential view first presented by Diodorus in the early Hellenistic period, and might even be said to have been the inventor of “possible worlds”. In this essay we take Scotus’ suggestion seriously devising first a double-index logic and then introducing the temporal order. Finally, using the temporal order, we (...) define a modal order. This allows us to present modal logic without the usual interpretive questions arising concerning the relation called variously ‘accessibility’, ‘alternativeness’, and, ‘relative possibility.’ The system in which this analysis is done is one of those which have come to be called a hybrid logic. (shrink)