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)

The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with (...) proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. (shrink)

A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving a representation theorem for 'combinatory posets.' A philosophical interpretation is given of the models, showing that an element of a combinatory poset can be understood simultaneously as a set of states and as a set of actions on states. This double interpretation allows for (...) one such element to be applied to another . Application turns out to be modeled the same way as 'fusion' in relevance logic.We also introduce 'dual combinators' that apply from the right. We then explore relationships to some well-known substructural logics, showing that each can be embedded into the structurally free. non-associative Lambek calculus, with the embedding taking a theorem ω to a statement of the form Γ ⊨ ω, where Γ is some fusion of the combinators needed to justify the structural assumptions of the given substructural logic. This builds on earlier ideas from Belnap and Meyer about a Gentzen system wherein structural rules are replaced with rules for introducing combinators. We develop such a system and prove a cut theorem. (shrink)

The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including some (...) new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. (shrink)