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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. Alan R. Anderson & Nuel D. Belnap (1975). Entailment: The Logic of Relevance and Neccessity, Vol. I. Princeton University Press.
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  3. Alan Ross Anderson (1990). Entailment: The Logic of Relevance and Necessity. Princeton University Press.
  4. Alan Anderson, Belnap R., D. Nuel & J. Michael Dunn (1992). Entailment: The Logic of Relevance and Necessity, Vol. Ii. Princeton University Press.
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  5. 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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  6. Arnon Avron (1992). Whither Relevance Logic? Journal of Philosophical Logic 21 (3):243 - 281.
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  7. 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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  8. 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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  9. Arnon Avron (1986). On Purely Relevant Logics. Notre Dame Journal of Formal Logic 27 (2):180-194.
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  10. Maria Baghramian (1988). The Justification for Relevance Logic. Philosophical Studies 32:32-43.
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  11. John A. Barker (1975). Relevance Logic, Classical Logic, and Disjunctive Syllogism. Philosophical Studies 27 (6):361 - 376.
  12. 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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  13. 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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  14. 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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  15. Jc Beall, Ross Brady, Michael Dunn, Allen Hazen, Edwin Mares, John Slaney, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley & 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. Ross T. Brady (1996). Gentzenizations of Relevant Logics with Distribution. Journal of Symbolic Logic 61 (2):402-420.
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  23. 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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  24. Ross T. Brady (1996). Gentzenizations of Relevant Logics Without Distribution. II. Journal of Symbolic Logic 61 (2):379-401.
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  25. 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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  26. Ross T. Brady (1992). Hierarchical Semantics for Relevant Logics. Journal of Philosophical Logic 21 (4):357 - 374.
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  27. 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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  28. 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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  29. 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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  30. M. W. Bunder (1979). A More Relevant Relevance Logic. Notre Dame Journal of Formal Logic 20 (3):701-704.
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  31. G. Charlwood (1981). An Axiomatic Version of Positive Semilattice Relevance Logic. Journal of Symbolic Logic 46 (2):233-239.
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  32. John Chidgey (1979). On the Non-Availability of Dawson-Modeling Into Certain Relevance Alethic Modal Logics. Studia Logica 38 (2):89 - 94.
    This paper shows that the Dawson technique of modelling deontic logics into alethic modal logics to gain insight into deontic formulas is not available for modelling a normal (in the spirit of Anderson) relevance deontic modal logic into either of the normal relevance alethic modal logics R S4or R M. The technique is to construct an extension of the well known entailment matrix set M 0and show that the model of the deontic formula P (A v B). PA v PB (...)
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  33. B. J. Copeland (1979). On When a Semantics is Not a Semantics: Some Reasons for Disliking the Routley-Meyer Semantics for Relevance Logic. [REVIEW] Journal of Philosophical Logic 8 (1):399 - 413.
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  34. Fabrice Correia (2004). Semantics for Analytic Containment. Studia Logica 77 (1):87 - 104.
    In 1977, R. B. Angell presented a logic for <span class='Hi'>analytic</span> containment, a notion of relevant implication stronger than Anderson and Belnap's entailment. In this paper I provide for the first time the logic of first degree <span class='Hi'>analytic</span> containment, as presented in [2] and [3], with a semantical characterization—leaving higher degree systems for future investigations. The semantical framework I introduce for this purpose involves a special sort of truth-predicates, which apply to pairs of collections of formulas instead of individual (...)
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  35. Jan Dejnožka (2010). The Concept of Relevance and the Logic Diagram Tradition. Logica Universalis 4 (1):67-135.
    What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...)
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  36. Harry Deutsch (1985). A Note on the Decidability of a Strong Relevant Logic. Studia Logica 44 (2):159 - 164.
    A modified filtrations argument is used to prove that the relevant logic S of [2] is decidable.
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  37. Eric Dietrich (2008). The Bishop and Priest: Toward a Point-of-View Based Epistemology of True Contradictions. Logos Architekton 2 (2):35-58..
    True contradictions are taken increasingly seriously by philosophers and logicians. Yet, the belief that contradictions are always false remains deeply intuitive. This paper confronts this belief head-on by explaining in detail how one specific contradiction is true. The contradiction in question derives from Priest's reworking of Berkeley's argument for idealism. However, technical aspects of the explanation offered here differ considerably from Priest's derivation. The explanation uses novel formal and epistemological tools to guide the reader through a valid argument with, not (...)
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  38. Kosta Došen (1992). The First Axiomatization of Relevant Logic. Journal of Philosophical Logic 21 (4):339 - 356.
    This is a review, with historical and critical comments, of a paper by I. E. Orlov from 1928, which gives the oldest known axiomatization of the implication-negation fragment of the relevant logic R. Orlov's paper also foreshadows the modal translation of systems with an intuitionistic negation into S4-type extensions of systems with a classical, involutive, negation. Orlov introduces the modal postulates of S4 before Becker, Lewis and Gödel. Orlov's work, which seems to be nearly completely ignored, is related to the (...)
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  39. Kosta Došen (1981). A Reduction of Classical Propositional Logic to the Conjunction-Negation Fragment of an Intuitionistic Relevant Logic. Journal of Philosophical Logic 10 (4):399 - 408.
  40. J. Michael Dunn (1987). Relevant Predication 1: The Formal Theory. [REVIEW] Journal of Philosophical Logic 16 (4):347 - 381.
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  41. J. Michael Dunn (1979). Relevant Robinson's Arithmetic. Studia Logica 38 (4):407 - 418.
    In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in (...)
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  42. J. Michael Dunn (1979). A Theorem in 3-Valued Model Theory with Connections to Number Theory, Type Theory, and Relevant Logic. Studia Logica 38 (2):149 - 169.
    Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...)
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  43. Michael Dunn & Greg Restall (2002). Relevance Logic. In D. Gabbay & F. Guenthner (eds.), Handbook of Philosophical Logic. Kluwer.
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  44. Kit Fine (1988). Semantics for Quantified Relevance Logic. Journal of Philosophical Logic 17 (1):27 - 59.
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  45. Bas C. Van Fraassen (1983). Gentlemen's Wagers: Relevant Logic and Probability. Philosophical Studies 43 (1):47 - 61.
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  46. James B. Freeman & Charles B. Daniels (1979). A Second-Order Relevance Logic with Modality. Studia Logica 38 (2):113 - 135.
    In this paper a system, RPF, of second-order relevance logic with S5 necessity is presented which contains a defined, notion of identity for propositions. A complete semantics is provided. It is shown that RPF allows for more than one necessary proposition. RPF contains primitive syntactic counterparts of the following semantic notions: (1) the reflexive, symmetrical, transitive binary alternativeness relation for S5 necessity, (2) the ternary Routley-Meyer alternativeness relation for implication, and (3) the Routley-Meyer notion of a prime intensional theory, as (...)
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  47. Harvey Friedman & Robert K. Meyer (1992). Whither Relevant Arithmetic? Journal of Symbolic Logic 57 (3):824-831.
    Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...)
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  48. André Fuhrmann (1990). Models for Relevant Modal Logics. Studia Logica 49 (4):501 - 514.
    Semantics are given for modal extensions of relevant logics based on the kind of frames introduced in [7]. By means of a simple recipe we may obtain from a class FRM (L) of unreduced frames characterising a (non-modal) logic L, frame-classes FRM (L.M) characterising conjunctively regular modal extensions L.M of L. By displaying an incompleteness phenomenon, it is shown how the recipe fails when reduced frames are under consideration.
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  49. Dov M. Gabbay & Ruy J. G. B. de Queiroz (1992). Extending the Curry-Howard Interpretation to Linear, Relevant and Other Resource Logics. Journal of Symbolic Logic 57 (4):1319-1365.
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  50. Dov Gabbay, Rolf Nossum & John Woods (2006). Context-Dependent Abduction and Relevance. Journal of Philosophical Logic 35 (1):65 - 81.
    Based on the premise that what is relevant, consistent, or true may change from context to context, a formal framework of relevance and context is proposed in which • contexts are mathematical entities • each context has its own language with relevant implication • the languages of distinct contexts are connected by embeddings • inter-context deduction is supported by bridge rules • databases are sets of formulae tagged with deductive histories and the contexts they belong to • abduction and revision (...)
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