About this topic

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 2007 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. The Philosophy of Alternative Logics.Andrew Aberdein & Stephen Read - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. pp. 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 (...)
  2. Anderson, W. The Cultivation of Whiteness (Anderson, Crotty, Garton, and Turnbull) 153 Abir-Am, P. And Elliott, C.(Eds) Commemorative Practices in Sciences Osiris Vol. 14 (Notice-NR) 139. [REVIEW]C. J. Acker, G. Baker, J. C. Beall, B. van Fraassen, K. Benson, P. Rehbock, F. Bevilacqua, E. Giannetto, M. Matthews & M. Boon - 2003 - Metascience 12:455-461.
  3. Lévy Azriel. On Ackermann's Set Theory.W. Ackermann - 1960 - Journal of Symbolic Logic 25 (4):355.
  4. Review: Azriel Levy, On Ackermann's Set Theory. [REVIEW]W. Ackermann - 1960 - Journal of Symbolic Logic 25 (4):355-355.
  5. Kripke Models for Linear Logic.Gerard Allwein & J. Michael Dunn - 1993 - 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, (...)
  6. Entailment: The Logic of Relevance and Neccessity, Vol. I.Alan R. Anderson & Nuel D. Belnap - 1975 - Princeton University Press.
  7. Entailment: The Logic of Relevance and Necessity.Alan Ross Anderson - 1975 - Princeton University Press.
  8. Completeness Theorems for the Systems E of Entailment and EQ of Entailment with Quantification.Alan Ross Anderson - 1960 - Mathematical Logic Quarterly 6 (7‐14):201-216.
  9. Completeness Theorems for the Systems E of Entailment and EQ of Entailment with Quantification.Alan Ross Anderson - 1960 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 6 (7-14):201-216.
  10. Entailment. Vol. 1.Alan Ross Anderson & Nuel D. Belnap - 1977 - Canadian Journal of Philosophy 7 (2):405-411.
  11. Tautological Entailments.Alan Ross Anderson & Nuel D. Belnap - 1962 - Philosophical Studies 13 (1-2):9 - 24.
  12. Entailment: The Logic of Relevance and Necessity, Vol. II.Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn - 1992 - Princeton University Press.
  13. Still Rainin' Still Dreamin': Hall Anderson's Ketchikan.Hall Anderson - 2010 - University of Alaska Press.
  14. Entailment and Modality.R. W. Ashby - 1962 - Proceedings of the Aristotelian Society 63:203 - 216.
  15. Implicational F-Structures and Implicational Relevance Logics.A. Avron - 2000 - 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 (...)
  16. Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening.A. Avron - 1998 - 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, (...)
  17. The Classical Constraint on Relevance.Arnon Avron - 2014 - Logica Universalis 8 (1):1-15.
    We show that as long as the propositional constants t and f are not included in the language, any language-preserving extension of any important fragment of the relevance logics R and RMI can have only classical tautologies as theorems . This property is not preserved, though, if either t or f is added to the language, or if the contraction axiom is deleted.
  18. What is Relevance Logic?Arnon Avron - 2014 - Annals of Pure and Applied Logic 165 (1):26-48.
    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, R → source is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second definition, while all fragments of linear logic are not.
  19. Whither Relevance Logic?Arnon Avron - 1992 - Journal of Philosophical Logic 21 (3):243 - 281.
  20. Relevance and Paraconsistency---A New Approach. II. The Formal Systems.Arnon Avron - 1990 - Notre Dame Journal of Formal Logic 31 (2):169-202.
  21. Relevance and Paraconsistency--A New Approach.Arnon Avron - 1990 - Journal of Symbolic Logic 55 (2):707-732.
  22. A Constructive Analysis of RM.Arnon Avron - 1987 - Journal of Symbolic Logic 52 (4):939 - 951.
  23. On an Implication Connective of RM.Arnon Avron - 1986 - Notre Dame Journal of Formal Logic 27 (2):201-209.
  24. On Purely Relevant Logics.Arnon Avron - 1986 - Notre Dame Journal of Formal Logic 27 (2):180-194.
  25. Relevant Entailment--Semantics and Formal Systems.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (2):334-342.
  26. Entailment and the Modal Fallacy.John Bacon - 1965 - Review of Metaphysics 18 (3):566 - 571.
  27. The Relevant Fragment of First Order Logic.Guillermo Badia - forthcoming - Review of Symbolic Logic:1-24.
    Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.
  28. A Lindström-Style Theorem for Finitary Propositional Weak Entailment Languages with Absurdity.Guillermo Badia - 2016 - Logic Journal of the IGPL 24 (2):115-137.
    Following a result by De Rijke for modal logic, it is shown that the basic weak entailment model-theoretic language with absurdity is the maximal model-theoretic language having the finite occurrence property, preservation under relevant directed bisimulations and the finite depth property. This can be seen as a generalized preservation theorem characterizing propositional weak entailment formulas among formulas of other model-theoretic languages.
  29. The Justification for Relevance Logic.Maria Baghramian - 1988 - Philosophical Studies 32:32-43.
  30. A New Approach to Classical Relevance.Inge Bal & Peter Verdée - 2015 - Studia Logica 103 (5):919-954.
    In this paper we present a logic that determines when implications in a classical logic context express a relevant connection between antecedent and consequent. In contrast with logics in the relevance logic literature, we leave classical negation intact—in the sense that the law of non-contradiction can be used to obtain relevant implications, as long as there is a connection between antecedent and consequent. On the other hand, we give up the requirement that our theory of relevance should be able to (...)
  31. Review: Paul Henle, Mysticism and Semantics. [REVIEW]Y. Bar-Hillel - 1966 - Journal of Symbolic Logic 31 (3):497-497.
  32. Review: R. M. Martin, J. H. Woodger, Toward an Inscriptional Semantics. [REVIEW]Y. Bar-Hillel - 1952 - Journal of Symbolic Logic 17 (1):71-72.
  33. Review: Alan Pasch, Experience and the Analytic. A Reconsideration of Empiricism. [REVIEW]Yehoshua Bar-Hillel - 1962 - Journal of Symbolic Logic 27 (2):221-221.
  34. Relevance Logic, Classical Logic, and Disjunctive Syllogism.John A. Barker - 1975 - Philosophical Studies 27 (6):361 - 376.
  35. Lógica positiva : plenitude, potencialidade e problemas (do pensar sem negação).Tomás Barrero - 2004 - 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 (...)
  36. Anderson, JR, 123 Arterberry, ME, 1 Aslin, RN, B33 Au, TK-F., B53.H. Barth, M. H. Bornstein, J. I. D. Campbell, B. Geurts, P. C. Gordon, R. Gunter, R. Hendrick, C. W. Hue, S. Laurence & E. Margolis - 2003 - Cognition 86:317.
  37. Natural Deduction for Non-Classical Logics.David Basin, Seán Matthews & Luca Viganò - 1998 - 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 (...)
  38. Propositional Logic Extended with a Pedagogically Useful Relevant Implication.Diderik Batens - 2014 - Logic and Logical Philosophy 23 (3).
    First and foremost, this paper concerns the combination of classical propositional logic with a relevant implication. The proposed combination is simple and transparent from a proof theoretic point of view and at the same time extremely useful for relating formal logic to natural language sentences. A specific system will be presented and studied, also from a semantic point of view. The last sections of the paper contain more general considerations on combining classical propositional logic with a relevant logic that has (...)
  39. A Dynamic Characterization of the Pure Logic of Relevant Implication.Diderik Batens - 2001 - 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 (...)
  40. Relevant Implication and the Weak Deduction Theorem.Diderik Batens - 1987 - 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.
  41. Relevant Derivability and Classical Derivability in Fitch-Style and Axiomatic Formulations of Relevant Logics.Diderik Batens & Jean Paul Van Bendegem - 1985 - Logique Et Analyse 109 (9):22-31.
  42. Review: Nuel D. Belnap, Entailment and Relevance. [REVIEW]A. Bayart - 1969 - Journal of Symbolic Logic 34 (1):120-120.
  43. Anderson Alan Ross and Belnap Nuel D. Jr., Modalities in Ackermann's “Rigorous Implication.” The Journal of Symbolic Logic, Vol. 24 No. 2 , Pp. 107–111. [REVIEW]A. Bayart - 1969 - Journal of Symbolic Logic 34 (1):120.
  44. Belnap Nuel D. Jr., Entailment and Relevance.A. Bayart - 1969 - Journal of Symbolic Logic 34 (1):120.
  45. Review: Alan Ross Anderson, Nuel D. Belnap, Modalities in Ackerman's Rigorous Implication. [REVIEW]A. Bayart - 1969 - Journal of Symbolic Logic 34 (1):120-120.
  46. Review: Alan H. Gardiner, The Theory of Proper Names. A Controversial Essay. [REVIEW]Charles A. Baylis - 1958 - Journal of Symbolic Logic 23 (2):212-212.
  47. Review: Alan Ross Anderson, A Note on Subjunctive and Counterfactual Conditionals. [REVIEW]Charles A. Baylis - 1953 - Journal of Symbolic Logic 18 (4):338-338.
  48. Review: C. Lewy, Entailment and Necessary Propositions. [REVIEW]Charles A. Baylis - 1951 - Journal of Symbolic Logic 16 (4):299-300.
  49. Review: P. F. Strawson, Necessary Propositions and Entailment-Statements. [REVIEW]Charles A. Baylis - 1949 - Journal of Symbolic Logic 14 (3):202-202.
  50. Review: S. Korner, On Entailment. [REVIEW]Charles A. Baylis - 1949 - Journal of Symbolic Logic 14 (3):199-200.
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