In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people working in (...) the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume. (shrink)

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of (...) these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa. (shrink)

This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the 'Australian Plan' semantics, which uses a unary operator '⋆' for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it provides a (...) semantics for the contraction-free relevant logic C (or RW). Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof. (shrink)