According to an intuitive picture of relevant entailment, an entailment is relevant if all the formulas it contains contribute to its validity. In this paper, we provide a ground-theoretic analysis of this notion of contribution, and as a result of relevant entailment. We build a system of bilateral logical grounding within which we can derive classical entailment and analyze the contribution of premises and conclusions, in terms of a certain type of connection between (...) their respective logical grounds. The resulting framework is shown to have a strong unifying and explanatory power by bringing together the notions of grounding and relevance and explaining the commonalities and differences between alternative contribution-based accounts of relevance. (shrink)

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)

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. Originally published in 1992. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

The title of this and proposed second volume presents the basic idea which unifies the wide variety of topics developed and investigated by the principal authors, major contributing authors, J. M. Dunn and Robert K. Meyer, and eleven other contributors. The other contributors are: J. R. Chidgey, J. A. Coffa, Dorthy L. Grover, Bas van Fraassen, H. Leblanc, Storrs McCall, A. Parks, G. Pottinger, R. Routley, A. Urquhart, and R. G. Wolf. From both the useful analytic table of contents and (...) from the section titles it is clear which contributing author has written each section. Chapter I, "The Pure Calculus of Entailment," suggests how logics of entailment are logics of relevance and necessity. When we explain why a large number of systems are investigated, it will be clear why we write of logics of entailment instead of the logic of entailment. The principal authors build a case in Chapter I, and elsewhere, that a genuinely valid formal inference, viz., a formal entailment, from a formula A to a formula B requires that A be relevant to B in addition to it being impossible that there be an interpretation making A true and B false. So, a claim that A entails B tells us, explicitly or implicitly, that A is relevant to B and that "if A then B" is necessarily true, i.e., A strictly implies B. Consequently since a logic of entailment presents allegedly unfalsifiable claims, i.e., theses, using entailment claims, a logic of entailment presents theses about relevance between antecedents and consequents and necessity. It should be noted that the systems of logic developed are for formal entailments; they may not be logics for material entailments such as: being a cube entails having six surfaces. (shrink)

This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc- tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven. -/- .

In this paper I undermine the Entailment Principle according to which if an entity is a truthmaker for a certain proposition and this proposition entails another, then the entity in question is a truthmaker for the latter proposition. I argue that the two most promising versions of the principle entail the popular but false Conjunction Thesis, namely that a truthmaker for a conjunction is a truthmaker for its conjuncts. One promising version of the principle understands entailment as strict (...) implication but restricts the field of application of the principle to purely contingent truths (i.e. those that contain no necessary proposition at any level of analysis). But a conjunction of purely contingent truths strictly implies its conjuncts. So this version of the principle is committed to the Conjunction Thesis. The same is true of the version of the principle where entailment is understood in the sense of systems T, R, and E of relevant logic, since in these systems conjunctions entail their conjuncts. I argue that the Conjunction Thesis is false because a truthmaker is that in virtue of what a certain proposition is true and it is false that, for example, what the proposition that Peter is a man is true in virtue of is the conjunctive fact that Peter is man and Saturn is a planet (or the facts that Peter is a man and that Saturn is a planet taken together). I also argue against other versions of the principle. (shrink)

Knowledge is widely thought to entail belief. But Radford has claimed to offer a counterexample: the case of the unconfident examinee. And Myers-Schulz and Schwitzgebel have claimed empirical vindication of Radford. We argue, in defense of orthodoxy, that the unconfident examinee does indeed have belief, in the epistemically relevant sense of dispositional belief. We buttress this with empirical results showing that when the dispositional conception of belief is specifically elicited, people’s intuitions then conform with the view that knowledge entails (...) (dispositional) belief. (shrink)

Graham Priest has asked whether the consequence relation associated with the Anderson–Belnap system of Tautological Entailment,1 in the language with connectives ¬, ∧, ∨, and countably many propositional variables as tomic formulas, maximal amongst the substitution-invariant relevant consequence relations on this language. Here a consequence relation is said to be relevant just in case whenever for a set of formulas Γ and formula B, we have Γ B only if some propositional variable occurring in B occurs in (...) at least one formula in Γ. (It follows that relevant consequence relations are atheorematic in the sense that whenever Γ B for some such consequence relation , Γ = ∅.) Here I write up in more detail the upshot of the conversation – returning an affirmative answer to Priest’s question – about this in the common room that Greg Restall and I were participating in last Friday [ = October 6, 2006], dotting some “i”s and crossing some “t”s (and adding the odd further reflection). (shrink)

A transcription of Richard Routley's manuscript, "Semantic Analysis of Entailment and Relevant Implication: I".

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)

In "The Logical Structure of Linguistic Commitment I" (The Journal of Philosophical Logic 23 (1994), 369-400), we sketch a linguistic theory (inspired by Brandom's Making it Explicit) which includes an "expressivist" account of the implication connective, →: the role of → is to "make explicit" the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A → B) as "commitment to A is, in part, commitment to B". Our project is to study (...) the logic of →. LSLC I approximates (A → B) as "anyone committed to A is committed to B", ignoring issues of whether A is relevant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine's semantics for relevance logics, a reading that Fine suggests. (shrink)

This book provides a new philosophical, semantical and historical analysis of and justification for the relevant logic of entailment. Its fresh and original perspective on the logic of entailment will be valuable for all who want to know more about the historical and philosophical origins of modern symbolic logic.

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary (...) property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing. (shrink)

In Categories 12 (14b11–22), Aristotle famously claims that [1] true sentences and reality stand in a mutually implicative relationship, and that [2] reality causes the truth of sentences but not vice versa. In this paper, I first argue that Boethius’ reading of the above passage led medieval logicians to assess [1] and [2] within the framework of a theory of consequence. Then, I consider two important questions raised by Boethius and later logicians in relation to [1] and [2], and, namely, (...) whether the implication described in [1] really satisfies the requirement for consequence, and whether the truthmaking relation posited in [2] can be boiled down to the relation of consequence. Boethius himself, I argue, answered affirmatively to both questions and thus (much like some contemporary truthmaker theorists) committed himself to an account of truthmaking as entailment. I then consider criticisms to Boethius’ account found in the writings of William of Champeaux, two anonymous authors connected to the school of Jocelyn of Soissons, Peter Abelard, and the author of the Ars Meliduna. Here, I argue that the emergence of a concern for relevance drives the need for abandoning Boethius’ account as well as his conception of truthmaking as entailment. (shrink)

I argue that there are Leibnizian-style cosmological arguments for the existence of God which start from very mild premises which affirm the mere possibility of a principle of sufficient reason. The utilization of such premises gives a great deal of plausibility to such types of argumentation. I spend the majority of the paper defending three major objections to such mild premises viz., a reductio argument from Peter van Inwagen and William Rowe, which proffers and defends the idea that a necessary (...) proposition cannot explain a contingent one. I, then, turn to an amelioration of the Rowe/van Inwagen argument which attempts to appeal to an entailment relation between explanans and explanandum that is fettered out in terms of relevance logic. Subsequent to dispelling with that worry, I tackle objections to the utilization of weak principles of sufficient reason that depend essentially upon agglomerative accounts of explanation. (shrink)

We generalize the notion ofconsequence relationstandard in abstract treatments of logic to accommodate intuitions ofrelevance. The guiding idea follows theuse criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each beusedin some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining betweenmultisets. We motivate and state basic definitions of relevant consequence relations, (...) both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases). (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion (...) of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

Once upon a time, modal logic was castigated because it ‘had no semantics.’ Kripke, Hintikka, Kanger, and others changed all that. In a similar way, when Relevant Logic was introduced by Anderson and Belnap, it too was castigated for ‘having no semantics.’ The present overview marks a culmination of that effort. The semantic approach described here brings together a number of hitherto disparate efforts to set out formal systems for logics of relevant implication and entailment. It also (...) makes clear (despite some of our hopes and utterances) that the One True Logic does not exist. This is as true for relevant logics as Kripke et al., showed it to be for modal logics. In both cases, subtle (and not so subtle) variations on semantical postulates produce different logics in the same family. The question of which semantical postulates are correct makes no sense without further context, i.e., the questioner needs to answer the question: Correct for what? The question that does remain is: What motivates the relevant family of logics? And this is the question that is the main job for this chapter to investigate. (shrink)

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference (...) from a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency. (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)

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the (...) neighborhood semantics meets the Routley-Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable. (shrink)

Social constructionist claims are surprising and interesting when they entail that presumably natural kinds are in fact socially constructed. The claims are interesting because of their theoretical and political importance. Authors like Díaz-León argue that constitutive social construction is more relevant for achieving social justice than causal social construction. This paper challenges this claim. Assuming there are socially salient groups that are discriminated against, the paper presents a dilemma: if there were no constitutively constructed social kinds, the causes of (...) the discrimination of existing social groups would have to be addressed, and understanding causal social construction would be relevant to achieve social justice. On the other hand, not all possible constitutively socially constructed kinds are actual social kinds. If an existing social group is constitutively constructed as a social kind K, the fact that it actually exists as a K has social causes. Again, causal social construction is relevant. The paper argues that (i) for any actual social kind X, if X is constitutively socially constructed as K, then it is also causally socially constructed; and (ii) causal social construction is at least as relevant as constitutive social construction for concerns of social justice. For illustration, I draw upon two phenomena that are presumed to contribute towards the discrimination of women: (i) the poor performance effects of stereotype threat, and (ii) the silencing effects of gendered language use. (shrink)

The first part of this paper argues that if Craver’s ([2007a], [2007b]) popular mutual manipulability account (MM) of mechanistic constitution is embedded within Woodward’s ([2003]) interventionist theory of causation--for which it is explicitly designed--it either undermines the mechanistic research paradigm by entailing that there do not exist relationships of constitutive relevance or it gives rise to the unwanted consequence that constitution is a form of causation. The second part shows how Woodward’s theory can be adapted in such a way that (...) (MM) neither undermines the mechanistic paradigm nor reduces constitution to causation. However, it turns out that this modified theoretical embedding of (MM) makes it impossible to produce empirical evidence for constitutive relations. The paper ends by suggesting an additional criterion, the fat-handedness criterion, which, when combined with (MM), generates indirect empirical evidence for constitutive relevance. (shrink)

There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and (...) logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by Białynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values. In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant. (shrink)