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Relevant logics are a group of logics which attempt to block irrelevant conclusions being drawn from a set of premises. The following inferences are all valid in classical logic, where A and B are any sentences whatsoever: from A to B → A, B → B and B ∨ ¬B; from ¬A to A→B; and from A ∧ ¬A to B. But if A and B are utterly irrelevant to one another, many feel reluctant to call these inferences acceptable. Similarly for the validity of the corresponding material implications, often called ‘paradoxes’ of material implication. Relevant logic can be seen as the attempt to avoid these ‘paradoxes’.

Key works Many trace the beginnings of relevant logic to Anderson & Belnap 1962Anderson & Belnap 1975 is a key early book-length exposition of relevant logics. Routley & Meyer 1972 and Routley & Meyer 1972 develop the relational ‘Routley-Meyer’ semantics for relevant implication, which has proved vital to the success of relevant logics. Read 1988 and Mares 2004 set out the philosophy of relevant logics. Brady 2006 contains much of Brady's work on relevant logics (which has been important throughout their development).  Restall 1995 explores using 4-valued semantics for relevant logics. 
Introductions Mares 2012 is a recent introduction to the area. Jago 2013 surveys some of the most important recent work (2003–13) in relevant logic. The chapter on relevant logic in Priest 2001 introduces the logical details in a concise way.
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  1. Andrew Aberdein & Stephen Read (2009). The Philosophy of Alternative Logics. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press 613-723.
    This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...)
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  2. Gerard Allwein & J. Michael Dunn (1993). Kripke Models for Linear Logic. Journal of Symbolic Logic 58 (2):514-545.
    We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, (...)
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  3. Alan R. Anderson & Nuel D. Belnap (1975). Entailment: The Logic of Relevance and Neccessity, Vol. I. Princeton University Press.
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  4. Alan Ross Anderson (1990). Entailment: The Logic of Relevance and Necessity. Princeton University Press.
  5. Alan Ross Anderson (1960). Completeness Theorems for the Systems E of Entailment and EQ of Entailment with Quantification. Mathematical Logic Quarterly 6 (7‐14):201-216.
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  6. Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton University Press.
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  7. A. Avron (2000). Implicational F-Structures and Implicational Relevance Logics. Journal of Symbolic Logic 65 (2):788-802.
    We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logic is that (...)
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  8. A. Avron (1998). Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening. Journal of Symbolic Logic 63 (3):831-859.
    We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...)
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  9. Arnon Avron (1992). Whither Relevance Logic? Journal of Philosophical Logic 21 (3):243 - 281.
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  10. Arnon Avron (1990). Relevance and Paraconsistency---A New Approach. II. The Formal Systems. Notre Dame Journal of Formal Logic 31 (2):169-202.
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  11. Arnon Avron (1990). Relevance and Paraconsistency---A New Approach. II. The Formal Systems. Notre Dame Journal of Formal Logic 31 (2):169-202.
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  12. Arnon Avron (1990). Relevance and Paraconsistency--A New Approach. Journal of Symbolic Logic 55 (2):707-732.
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  13. Arnon Avron (1987). A Constructive Analysis of RM. Journal of Symbolic Logic 52 (4):939 - 951.
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  14. Arnon Avron (1986). On an Implication Connective of RM. Notre Dame Journal of Formal Logic 27 (2):201-209.
  15. Arnon Avron (1986). On Purely Relevant Logics. Notre Dame Journal of Formal Logic 27 (2):180-194.
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  16. Arnon Avron (1984). Relevant Entailment--Semantics and Formal Systems. Journal of Symbolic Logic 49 (2):334-342.
  17. Maria Baghramian (1988). The Justification for Relevance Logic. Philosophical Studies 32:32-43.
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  18. Richard J. Baldauf (1984). Keep It Relevant! BioScience 34 (1):4-4.
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  19. John A. Barker (1975). Relevance Logic, Classical Logic, and Disjunctive Syllogism. Philosophical Studies 27 (6):361 - 376.
  20. Tomás Barrero (2004). Lógica positiva : plenitude, potencialidade e problemas (do pensar sem negação). Dissertation, Universidade Estadual de Campinas
    This work studies some problems connected to the role of negation in logic, treating the positive fragments of propositional calculus in order to deal with two main questions: the proof of the completeness theorems in systems lacking negation, and the puzzle raised by positive paradoxes like the well-known argument of Haskel Curry. We study the constructive com- pleteness method proposed by Leon Henkin for classical fragments endowed with implication, and advance some reasons explaining what makes difficult to extend this constructive (...)
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  21. David Basin, Seán Matthews & Luca Viganò (1998). Natural Deduction for Non-Classical Logics. Studia Logica 60 (1):119-160.
    We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness and (...)
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  22. Diderik Batens (2001). A Dynamic Characterization of the Pure Logic of Relevant Implication. Journal of Philosophical Logic 30 (3):267-280.
    This paper spells out a dynamic proof format for the pure logic of relevant implication. (A proof is dynamic if a formula derived at some stage need not be derived at a later stage.) The paper illustrates three interesting points. (i) A set of properties that characterizes an inference relation on the (very natural) dynamic proof interpretation, need not characterize the same inference relation (or even any inference relation) on the usual settheoretical interpretation. (ii) A proof format may display an (...)
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  23. Diderik Batens (1987). Relevant Implication and the Weak Deduction Theorem. Studia Logica 46 (3):239 - 245.
    It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.
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  24. Jc Beall (2014). Eco-Logical Lives: The Philosophical Lives of Richard Routley/Sylvan and Val Routley/Plumwood, by Dominic Hyde. Australasian Journal of Philosophy 93 (3):619-621.
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  25. Jc Beall (2011). Adding to Relevant Restricted Quantification. Australasian Journal of Philosophy 10:36-44.
    This paper presents, in a more general setting, a simple approach to ‘relevant restricted generalizations’ advanced in previous work. After reviewing some desiderata for restricted generalizations, I present the target route towards achieving the desiderata. An objection to the approach, due to David Ripley, is presented, followed by three brief replies, one from a dialetheic perspective and the others more general.
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  26. JC Beall, Ross T. Brady, A. P. Hazen, Graham Priest & Greg Restall (2006). Relevant Restricted Quantification. Journal of Philosophical Logic 35 (6):587 - 598.
    The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.
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  27. Jc Beall, Ross Brady, J. Michael Dunn, A. P. Hazen, Edwin Mares, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley, John Slaney & Richard Sylvan (2012). On the Ternary Relation and Conditionality. Journal of Philosophical Logic 41 (3):595 - 612.
    One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...)
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  28. Nuel D. Belnap, Anil Gupta & J. Michael Dunn (1980). A Consecutive Calculus for Positive Relevant Implication with Necessity. Journal of Philosophical Logic 9 (4):343-362.
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  29. Francesco Berto (2012). Non-Normal Worlds and Representation. In Michal Peliš & Vít Punčochář (eds.), The Logica Yearbook. College Publications
    World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic (...)
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  30. Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.) (2013). Paraconsistency: Logic and Applications. Springer.
    A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...)
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  31. Katalin Bimbó, J. Michael Dunn & Roger D. Maddux (2009). Relevance Logics and Relation Algebras. Review of Symbolic Logic 2 (1):102-131.
    Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...)
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  32. W. J. Blok & J. G. Raftery (2004). Fragments of R-Mingle. Studia Logica 78 (1-2):59 - 106.
    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 (...)
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  33. Susanne Bobzien (1999). Logic: The Stoics (Part Two). In Keimpe Algra, Jonathan Barnes & et al (eds.), The Cambridge History of Hellenistic Philosophy. CUP
    ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of deduction can be found (...)
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  34. Susanne Bobzien (1996). Stoic Syllogistic. Oxford Studies in Ancient Philosophy 14:133-92.
    ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...)
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  35. R. T. Brady (1984). Natural deduction systems for some quantified relevant logics. Logique Et Analyse 27 (8):355.
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  36. R. Brady & Nicholas Griffin (2005). REVIEWS-Relevant Logics and Their Rivals, Volume II. Bulletin of Symbolic Logic 11 (1):70-71.
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  37. Ross Brady (2008). Negation in Metacomplete Relevant Logics. Logique Et Analyse 51.
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  38. Ross T. Brady (1996). Gentzenizations of Relevant Logics with Distribution. Journal of Symbolic Logic 61 (2):402-420.
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  39. Ross T. Brady (1996). Relevant Implication and the Case for a Weaker Logic. Journal of Philosophical Logic 25 (2):151 - 183.
    We collect together some misgivings about the logic R of relevant inplication, and then give support to a weak entailment logic $DJ^{d}$ . The misgivings centre on some recent negative results concerning R, the conceptual vacuousness of relevant implication, and the treatment of classical logic. We then rectify this situation by introducing an entailment logic based on meaning containment, rather than meaning connection, which has a better relationship with classical logic. Soundness and completeness results are proved for $DJ^{d}$ with respect (...)
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  40. Ross T. Brady (1996). Gentzenizations of Relevant Logics Without Distribution. I. Journal of Symbolic Logic 61 (2):353-378.
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  41. Ross T. Brady (1996). Gentzenizations of Relevant Logics Without Distribution. II. Journal of Symbolic Logic 61 (2):379-401.
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  42. Ross T. Brady (1993). Rules in Relevant Logic — II: Formula Representation. Studia Logica 52 (4):565 - 585.
    This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A 1 .A 2 . ... .A n B, the conjunctive form,A 1&A 2 & ...A n (...)
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  43. Ross T. Brady (1992). Hierarchical Semantics for Relevant Logics. Journal of Philosophical Logic 21 (4):357 - 374.
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  44. Ross T. Brady (1991). Gentzenization and Decidability of Some Contraction-Less Relevant Logics. Journal of Philosophical Logic 20 (1):97 - 117.
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  45. Ross T. Brady (1989). A Content Semantics for Quantified Relevant Logics. II. Studia Logica 48 (2):243 - 257.
    In part I, we presented an algebraic-style of semantics, which we called “content semantics,” for quantified relevant logics based on the weak systemBBQ. We showed soundness and completeness with respect to theunreduced semantics ofBBQ. In part II, we proceed to show soundness and completeness for extensions ofBBQ with respect to this type of semantics. We introducereduced semantics which requires additional postulates for primeness and saturation. We then conclude by showing soundness and completeness forBB d Q and its extentions with respect (...)
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  46. Ross T. Brady (1989). A Routley-Meyer Affixing Style Semantics for Logics Containing Aristotle's Thesis. Studia Logica 48 (2):235 - 241.
    We provide a semantics for relevant logics with addition of Aristotle's Thesis, ∼(A→∼A) and also Boethius,(A→B)→∼(A→∼B). We adopt the Routley-Meyer affixing style of semantics but include in the model structures a regulatory structure for all interpretations of formulae, with a view to obtaining a lessad hoc semantics than those previously given for such logics. Soundness and completeness are proved, and in the completeness proof, a new corollary to the Priming Lemma is introduced (c.f.Relevant Logics and their Rivals I, Ridgeview, 1982).
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  47. Ross T. Brady (1984). Depth Relevance of Some Paraconsistent Logics. Studia Logica 43 (1-2):63 - 73.
    The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested ''s (...)
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  48. Ross Thomas Brady (2010). Free Semantics. Journal of Philosophical Logic 39 (5):511 - 529.
    Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic (...)
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  49. R. B. Braithwaite, Bertrade Russell & Friedrich Waismann (1938). The Relevance of Psychology to Logic. Aristotelian Society Supplementary Volume 17:19-68.
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  50. Andrew Brennan, Necessary and Sufficient Conditions. Stanford Encyclopedia of Philosophy.
    Describes the received theory of necessary and sufficient conditions, explains some standard objections to it, and lays out alternative ways of thinking about conditions and conditionals.
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