Search

The logic E4 is related to Brady's BN4 in a similar way to which Anderson and Belnap's logic of entailment E is related to their logic of the relevant implication R. In 'A companion to Brady's 4-valued relevant logic: the 4-valued logic of entailment E4', quoted in this paper, three alternatives to BN4 and another three to E4 are summarily introduced in a couple of pages as the only alternatives containing Routley and Meyer's basic logic B, provided some conditions are fulfilled. The aim of this note is to prove (i) BN4 and its three alternatives are the same logic up to some point, since they are functionally equivalent to each other, as it is the case with E4 and its three alternatives, which are also functionally equivalent to each other; (ii) to some extent, E4 is superior to BN4, since the latter is functionally included in the former, but not conversely. [ABSTRACT FROM AUTHOR]

This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

Copyright of Análisis Filosófico is the property of Sociedad Argentina de Analisis Filosofico and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The basic quasi-Boolean negation (QB-negation) expansions of relevance logics included in Anderson and Belnap's relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational semantics. [ABSTRACT FROM AUTHOR]

In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature (see, e.g. Mares, Studia Logica 51(1), 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71(1), 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are (Fine, Journal of Philosophical Logic 17(1), 27–59, 1988; Mares, Studia Logica 51(1), 1–20, 1992), both of which are quite dense and technically challenging. Second, there are a number of prima facie reasons to be worried about stratified semantics. The purpose of this paper is to revitalize research on stratified semantics. I will do so by giving a 'user friendly' presentation of the semantics, and by giving reasons to think that the prima facie reasons to be worried about it are too simplistic. [ABSTRACT FROM AUTHOR]

A logic can have certain desirable properties, but at the same time, it can exhibit some unattractive features too. We scrutinize some relevance logics--starting with implicational ticket entailment and continuing toward relevant implication with negation--from the point of view of the standard relational semantics for relevance logics. We define sound and complete semantics for the implicational fragments using a generalized accessibility relation and amending an earlier idea about the canonical situations. In some other cases, the semantics do not work out as expected. All the logics that we consider are decidable, which we use to prove that certain semantical constructions cannot come to fruition. [ABSTRACT FROM AUTHOR]

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic. [ABSTRACT FROM AUTHOR]

We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions (terms, sequents, equations, quasi-equations) in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics. [ABSTRACT FROM AUTHOR]

Abstract: We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, (the intensional fragment of R) is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second (more complicated) definition, while all fragments of linear logic are not. [Copyright &y& Elsevier]

The implicational fragment of the logic of relevant implication. R→ is knosvn to be decidable We show that the implicational fragment of the logic of ticket entailment. T→ is decidable. Oar proof is based on the raosecntiott calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of T→ to the decidability problem of R→. The decidability of T→ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base {B, B', I, W}. [ABSTRACT FROM AUTHOR]

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $$ LK $$ without any use of mix. [ABSTRACT FROM AUTHOR]

In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and refine these results to give an O( n log n)-space decision procedure for the basic positive relevance logic B+. We compute this upper-bound by first giving a sound and complete, cut-free, labelled sequent system for B+, and then establishing bounds on the application of the rules of this system. [ABSTRACT FROM AUTHOR]