This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is (...) also discussed. (shrink)

This paper surveys important work done in relevant logic in the past 10 years.

In many actual arguments, the conclusion seems intuitively to follow from the premisses, even though we cannot show that it follows logically. The traditional approach to evaluating such arguments is to suppose that they have an unstated premiss whose explicit addition will produce an argument where the conclusion does follow logically. But there are good reasons for doubting that people so frequently leave the premisses of their arguments unstated. The inclination to suppose that they do stems from the belief that (...) the only way in which an argument's conclusion can follow definitely from its premisses is to follow logically. I argue that this belief is mistaken. I propose a revision of the current generic conception of logical consequence, and its variant specifications, to avoid the paradoxes of strict implication. The revised conception can then be naturally extended to include also what we might call 'enthymematic consequence'. This concept is a kind of consequence, whose properties merit investigation. (shrink)

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered are (...) semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras. (shrink)

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics (...) which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence. (shrink)

Priest argued in Fusion and Confusion (Priest in Topoi 34(1):55–61, 2015a) for a new concept of logical consequence over the relevant logic B, one where premises my be “confused” together. This paper develops Priest’s idea. Whereas Priest uses a substructural proof calculus, this paper provides a Hilbert proof calculus for it. Using this it is shown that Priest’s consequence relation is weaker than the standard Hilbert consequence relation for B, but strictly stronger than Anderson and Belnap’s original relevant notion of (...) consequence. Unlike the latter, however, Priest’s consequence relation does not satisfy a variant of the variable sharing property. This paper shows that how it can be modified so as to do so. Priest’s consequence relation turns out to be surprisingly weak in some respects. The prospects of strengthening it is raised and discussed in a broader philosophical context. (shrink)

Val Plumwood and Richard Sylvan argued from their joint paper The Semantics of First Degree Entailment and onward that the variable sharing property is but a mere consequence of a good entailment relation, indeed they viewed it as a mere negative test of adequacy of such a relation, the property itself being a rather philosophically barren concept. Such a relation is rather to be analyzed as a sufficiency relation free of any form of premise suppression. Suppression of premises, therefore, gained (...) center stage. Despite this, however, no serious attempt was ever made at analyzing the concept. This paper shows that their suggestions for how to understand it, either as the Anti-Suppression Principle or as the Joint Force Principle, turn out to yield properties strictly weaker than that of variable sharing. A suggestion for how to understand some of their use of the notion of suppression which clearly is not in line with these two mentioned principles is given, and their arguments to the effect that the Anderson and Belnap logics T, E and R are suppressive are shown to be both technically and philosophically wanting. Suppression-freedom, it is argued, cannot do the job Plumwood and Sylvan intended it to do. (shrink)

ABSTRACT In this paper, I argue that a number of influential Millian responses to Frege’s puzzle, which consist in denying that Frege’s data apply to natural languages, are not viable if logic is to play its role in legitimizing the logical appraisal of rational subjects. A notion of validity which does justice to the normativity of logic must make room for a distinction between valid inferences and enthymemes. I discuss the prospects of formal, relevant and manifest validity as candidates for (...) a notion which complies with this desideratum. Their success, or failure is argued to hang on the viability of a semantical account of de jure co-reference, which is in tension with standard Millian tenets. I conclude that these Millian theories face the following dilemma: either accept that there is no notion of logical validity which makes logic normative for reasoning, thus jeopardizing our well entrenched practices of rational appraisal; or accept that de jure co-reference is a real semantical relation. (shrink)

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociski, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations. (shrink)

Expressing a widely-held view, David Hitchcock claims that "an enthymematic argument ... assumes at least the truth of the argument's associated conditional ... whose antecedent is the conjunction of the argument's explicit premises and whose consequent is the argument's conclusion." But even definitionally, this view is problematic, since an argument's being "enthymematic" or incomplete with respect to its explicit premises means that the conclusion is not implied by these premises alone. The paper attempts to specify the ways in which the (...) view is incorrect, as well as seemingly correct (e.g., the case of a Modus Ponens wherein the major premise is implicit). -/- . (shrink)