Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In (...) this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature. (shrink)

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)

In this paper we look at two classic methods of deriving consequences from inconsistent premises: Rescher-Manor and Schotch-Jennings. The overall goal of the project is to confine the method of drawing consequences from inconsistent sets to those that do not require reference to any information outside of very general facts about the set of premises. Methods in belief revision often require imposing assumptions on premises, e.g., which are the important premises, how the premises relate in non-logical ways. Such assumptions enable (...) one to select a reasonable collection of formulas from all the formulas of the language. Basic versions of the classic methods only use logical relations between the premises. We compare and criticize each of the classic views with an eye to combining the views to get the most out of inconsistent premises. We do this in a way that will respect the theoretical grounding of each view while meeting our other restrictions. (shrink)

A certain approach to paraconsistency was initiated by works of R. Jennings and P. Schotch. In their “Inference and necessity” [4] they proposed a notion of a level of inconsistency of a given set of premises. This level is a measure that assigns to a given set of premises X, the least number of elements of covers of X that consist of consistent subsets of X. The idea of the level of inconsistency allows to formulate a paraconsistent inference relation called (...) by the authors forcing, while the obtained approach  preservationism. Similarly as classical inference relation is truth-preserving, the obtained inference relation is preserving the level of inconsistency. We will discuss some examples of inferences that are valid in the sense of Jennings-Schotch inference relation and rise some questions on them. Based on that we formulate an inference relation as an answer to the mentioned doubts. As regards forcing inference relation, the set of premises needed to derive a given conclusion can vary when changing covers from one to another. Our proposal is to stipulate to have some common set of relevant premises. (shrink)