Relevant and substructural logics

Publisher Summary This chapter discusses proof theory of relevant and substructural logics, and the model theory of these logics. The discipline of relevant logic grew out of an attempt to understand notions of consequence and conditionality where the conclusion of a valid argument is relevant to the premises, and where the consequent of a true conditional is relevant to the antecedent. The structural rules dictate admissible forms of transformations of premises contained in proofs. The chapter discusses how relevant logics are naturally counted as substructural logics, as certain commonly admitted structural rules are responsible for introducing irrelevant consequences into proofs.

[1]  Nuel Belnap,et al.  Linear Logic Displayed , 1989, Notre Dame J. Formal Log..

[2]  Alasdair Urquhart,et al.  Failure of interpolation in relevant logics , 1993, J. Philos. Log..

[3]  William C. Frederick,et al.  A Combinatory Logic , 1995 .

[4]  J. Michael Dunn,et al.  Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation, Implication, and Various Logical Operations , 1990, JELIA.

[5]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[6]  A. N. Prior,et al.  The Theory of Implication , 1963 .

[7]  Greg Restall.  A Note on Naive Set Theory in LP , 1992, Notre Dame J. Formal Log..

[8]  Sulla Derivabilita,et al.  RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO , 2008 .

[9]  Nuel D. Belnap,et al.  How a Computer Should Think , 2019, New Essays on Belnap-­Dunn Logic.

[10]  Greg Restall,et al.  A Useful Substructural Logic , 1994, Bull. IGPL.

[11]  E. Orlowska.  Relational interpretation of modal logics , 1988 .

[12]  Robert K. Meyer,et al.  The semantics ofR4 , 1993, J. Philos. Log..

[13]  John L. Bell,et al.  A New Approach to Quantum Logic , 1986, The British Journal for the Philosophy of Science.

[14]  G. Grätzer.  General Lattice Theory , 1978 .

[15]  R. Meyer,et al.  The semantics of entailment — III , 1973 .

[16]  Alasdair Urquhart,et al.  Semantics for relevant logics , 1972, Journal of Symbolic Logic.

[17]  Peter W. O'Hearn,et al.  The Logic of Bunched Implications , 1999, Bulletin of Symbolic Logic.

[18]  H. A. Lewis,et al.  ENTAILMENT: The Logic of Relevance and Necessity (Volume I) , 1978 .

[19]  Vaughan R. Pratt,et al.  Dynamic algebras as a well-behaved fragment of relation algebras , 1988, Algebraic Logic and Universal Algebra in Computer Science.

[20]  J. Beall,et al.  The Law of Non-Contradiction , 2004 .

[21]  Alfred Tarski,et al.  Relational selves as self-affirmational resources , 2008 .

[22]  Didier Galmiche.  Connection methods in linear logic and proof nets construction , 2000, Theor. Comput. Sci..

[23]  S. Gottwald.  Many-Valued Logics , 2007 .

[24]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[25]  Ross T. Brady,et al.  A content semantics for quantified relevant logics. II , 1988, Stud Logica.

[26]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .

[27]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[28]  Silvio Ghilardi,et al.  Constructive Canonicity in Non-Classical Logics , 1997, Ann. Pure Appl. Log..

[29]  Arnon Avron,et al.  Whither relevance logic? , 1992, J. Philos. Log..

[30]  Dock Bumpers,et al.  Volume 2 , 2005, Proceedings of the Ninth International Conference on Computer Supported Cooperative Work in Design, 2005..

[31]  Richard Routley,et al.  Relevant logics and their rivals , 1982 .

[32]  Chrysafis Hartonas,et al.  Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics , 1997, Stud Logica.

[33]  Greg Restall,et al.  Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star) , 1999 .

[34]  R. Seely,et al.  Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. , 1997 .

[35]  Richard Sylvan,et al.  The semantics of entailment—II , 1972, Journal of Philosophical Logic.

[36]  Hugues Leblanc.  Truth, syntax and modality : proceedings of the Temple University Conference on Alternative Semantics , 1973 .

[37]  J. Dunn.  Partial-Gaggles Applied to Logics with Restricted Structural Rules , 1991 .

[38]  Robert K. Meyer,et al.  Whither Relevant Arithmetic? , 1992, J. Symb. Log..

[39]  John L. Bell,et al.  Models and Ultraproducts: An Introduction. , 1969 .

[40]  Michael A. McRobbie,et al.  Automated theorem proving for non-classical logics , 1988, Research Notes in Theoretical Computer Science.

[41]  Nuel Belnap,et al.  Linear Analytic Tableaux , 1995, TABLEAUX.

[42]  J. Dunn,et al.  Curry's paradox , 1979 .

[43]  Graham Priest,et al.  Sociative Logics and Their Applications Essays , 2000 .

[44]  Haskell B. Curry,et al.  A Theory Of Formal Deducibility , 1950 .

[45]  Robert K. Meyer,et al.  Semantics of Entailment 0 , 1993 .

[46]  Richard Routley,et al.  Classical relevant logics II , 1973 .

[47]  Ludomir Newelski,et al.  Logic Colloquium 2005 (Lecture Notes in Logic) , 2007 .

[48]  G. Ryle,et al.  Contemporary aspects of philosophy , 1976 .

[49]  A. Heyting,et al.  Intuitionism: An introduction , 1956 .

[50]  Philip Kremer,et al.  Dunn's Relevant Predication, Real Properties and Identity , 1997 .

[51]  S. Crawford,et al.  Volume 1 , 2012, Journal of Diabetes Investigation.

[52]  J. Girard.  Geometry of interaction III: accommodating the additives , 1995 .

[53]  Garrel Pottinger.  On analysing relevance constructively , 1979 .

[54]  Graham Priest,et al.  Sense, entailment and Modus ponens , 1980, J. Philos. Log..

[55]  J. Michael Dunn,et al.  Relevant predication 1: The formal theory , 1987, J. Philos. Log..

[56]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[57]  Wojciech A. Trybulec.  Partially Ordered Sets , 1990 .

[58]  Robert K. Meyer,et al.  The Admissibility of γ in R4 , 1992, Notre Dame J. Formal Log..

[59]  H. Gaifman,et al.  Symbolic Logic , 1881, Nature.

[60]  D. Gabbay,et al.  Information Flow and the Lambek Calculus , 1994 .

[61]  Robert K. Meyer.  RI the Bounds of Finitude , 1970 .

[62]  F. J. Pelletier,et al.  316 Notre Dame Journal of Formal Logic , 1982 .

[63]  J. Girard.  Proof Theory and Logical Complexity , 1989 .

[64]  Philip Wadler,et al.  A Syntax for Linear Logic , 1993, MFPS.

[65]  maarten marx,et al.  Arrow logic and multi-modal logic , 1997 .

[66]  Alasdair Urquhart,et al.  Duality for algebras of relevant logics , 1996, Stud Logica.

[67]  Katalin Bimbó.  Semantics for Structurally Free Logics LC+ , 2001, Log. J. IGPL.

[68]  Andre Scedrov,et al.  Electronic Notes in Theoretical Computer Science: Preface , 1996 .

[69]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[70]  Kosta Dosen,et al.  Sequent-systems and groupoid models. I , 1988, Stud Logica.

[71]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[72]  Chrysafis Hartonas,et al.  Stone duality for lattices , 1997 .

[73]  Dov M. Gabbay,et al.  Investigations in modal and tense logics with applications to problems in philosophy and linguistics , 1976 .

[74]  American Journal of Mathematics , 1886, Nature.

[75]  G. Mints,et al.  Closed categories and the theory of proofs , 1981 .

[76]  Greg Restall,et al.  Display Logic and Gaggle Theory , 1995, Reports Math. Log..

[77]  William P. R. Mitchell.  The Carcinogenic Example , 1997, Log. J. IGPL.

[78]  Peter Hilton,et al.  Category Theory, Homology Theory and their Applications II , 1969 .

[79]  Michael Dummett,et al.  The logical basis of metaphysics , 1991 .

[80]  J. Girard,et al.  Proofs and types , 1989 .

[81]  Robert K. Meyer,et al.  ${\rm E}$ and ${\rm S}4$. , 1970 .

[82]  J. Girard,et al.  Advances in Linear Logic , 1995 .

[83]  Philip Wadler,et al.  A Taste of Linear Logic , 1993, MFCS.

[84]  Robert K. Meyer.  Metacompleteness , 1976, Notre Dame J. Formal Log..

[85]  Heinrich Wansing,et al.  Sequent Calculi for Normal Modal Proposisional Logics , 1994, J. Log. Comput..

[86]  Atwell R. Turquette,et al.  On the Many-Valued Logics , 1941 .

[87]  André Fuhrmann,et al.  A relevant theory of conditionals , 1995, J. Philos. Log..

[88]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[89]  Ross T. Brady.  A content semantics for quantified relevant logics. I , 1988, Stud Logica.

[90]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[91]  H. Wansing.  Displaying Modal Logic , 1998 .

[92]  J.F.A.K. van Benthem,et al.  Language in Action: Categories, Lambdas and Dynamic Logic , 1997 .

[93]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

[94]  John C. Mitchell,et al.  Foundations for programming languages , 1996, Foundation of computing series.

[95]  T. Schmidt-Kaler,et al.  An addendum to the , 2001 .

[96]  Richard Routley,et al.  Classical relevant logics. I , 1973 .

[97]  Leonard Lipshitz.  The undecidability of the word problems for projective geometries and modular lattices , 1974 .

[98]  J. Dunn,et al.  Relevant Predication 3: Essential Properties , 1990 .

[99]  H. Keisler,et al.  The Kleene Symposium , 1980 .

[100]  John Slaney.  A general logic , 1990 .

[101]  A. Troelstra.  Lectures on linear logic , 1992 .

[102]  Paul C. Gilmore.  Natural Deduction Based Set Theories: A New Resolution of the Old Paradoxes , 1986, J. Symb. Log..

[103]  Melvin Fitting,et al.  Bilattices and the Semantics of Logic Programming , 1991, J. Log. Program..

[104]  R. Meyer,et al.  Algebraic analysis of entailment I , 1972 .

[105]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[106]  Augustus de Morgan,et al.  On the Syllogism, No. IV., and on the Logic of Relations , 1864 .

[107]  II. Mathematisches.  Power and Weakness of the Modal Display Calculus , 1996 .

[108]  R. Blute,et al.  Natural deduction and coherence for weakly distributive categories , 1996 .

[109]  Kosta Dosen,et al.  Deductive completeness , 1996, Bull. Symb. Log..

[110]  A. Church.  Special Cases of the Decision Problem , 1951 .

[111]  Robert K. Meyer,et al.  Logic on the Australian plan , 1986, J. Philos. Log..

[112]  Jack Kaminsky.  Making it Explicit , 1999 .

[113]  Nuel Belnap,et al.  Gupta's rule of revision theory of truth , 1982, J. Philos. Log..

[114]  Robert K. Meyer,et al.  Combinators and Structurally Free Logic , 1997, Log. J. IGPL.

[115]  J. Dunn,et al.  Relevant predication 2: Intrinsic properties and internal relations , 1990 .

[116]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[117]  Mariangiola Dezani-Ciancaglini,et al.  Functional Characters of Solvable Terms , 1981, Math. Log. Q..

[118]  Y. Bar-Hillel.  A Quasi-Arithmetical Notation for Syntactic Description , 1953 .

[119]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[120]  Richard Routley,et al.  The Semantics of Entailment. , 1977 .

[121]  Robert K. Meyer,et al.  Independent Axioms for the Implicational Fragment of Sobociński's Three‐Valued Logic , 1972 .

[122]  B. Jack Copeland,et al.  On when a semantics is not a semantics: Some reasons for disliking the Routley-Meyer semantics for relevance logic , 1979, J. Philos. Log..

[123]  Arnon Avron.  On purely relevant logics , 1986, Notre Dame J. Formal Log..

[124]  Stephen Cole Kleene.  Disjunction and Existence Under Implication in Elementary Intuitionistic Formalisms , 1962, J. Symb. Log..

[125]  Robert K. Meyer,et al.  Solution to the P − W problem , 1982, Journal of Symbolic Logic.

[126]  Alasdair Urquhart,et al.  The complexity of decision procedures in relevance logic II , 1999, Journal of Symbolic Logic.

[127]  J. Dunn,et al.  Star and Perp: Two Treatments of Negation1 , 1993 .

[128]  Peter Aczel,et al.  Non-well-founded sets , 1988, CSLI lecture notes series.

[129]  G. Mints,et al.  Cut-elimination theorem for relevant logics , 1976 .

[130]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[131]  A. G. Dragálin.  Mathematical Intuitionism. Introduction to Proof Theory , 1988 .

[132]  Betti Venneri,et al.  Intersection Types as Logical Formulae , 1994, J. Log. Comput..

[133]  Heinrich Wansing,et al.  Proof Theory of Modal Logic , 1996 .

[134]  J. Michael Dunn.  Is Existence a (Relevant) Predicate , 1996 .

[135]  B. J. Copeland.  What is a Semantics for Classical Negation , 1986 .

[136]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[137]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[138]  Graham Priest,et al.  Paraconsistent Logic: Essays on the Inconsistent , 1990 .

[139]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic , 2002 .

[140]  Steve Giambrone.  TW+ and RW+ are decidable , 1985, J. Philos. Log..

[141]  Robert K. Meyer.  E and S4 , 1970, Notre Dame J. Formal Log..

[142]  I. Levi,et al.  Making It Explicit , 1994 .

[143]  A. N. Prior,et al.  Notes on the axiomatics of the propositional calculus , 1963, Notre Dame J. Formal Log..

[144]  Ross T. Brady.  Gentzenization and decidability of some contraction-less relevant logics , 1991, J. Philos. Log..

[145]  Valeria C V de Paiva,et al.  Term Assignment for Intuitionistic Linear Logic , 1992 .

[146]  Rajeev Goré,et al.  Substructural Logics on Display , 1998, Log. J. IGPL.

[147]  G. Charlwood,et al.  An axiomatic version of positive semilattice relevance logic , 1981, Journal of Symbolic Logic.

[148]  R. Goldblatt.  Mathematics of modality , 1993 .

[149]  Vincent Danos,et al.  On the linear decoration of intuitionistic derivations , 1995, Arch. Math. Log..

[150]  G. P. Henderson,et al.  Translations from the philosophical writings of Gottlob Frege , 1953 .

[151]  Edwin D. Mares,et al.  Relevant logic and the theory of information , 1996, Synthese.

[152]  Katalin Bimbó,et al.  Two Extensions of the Structurally Free Logic LC* , 1998, Log. J. IGPL.

[153]  Chrysafis Hartonas,et al.  Order-Duality, Negation and Lattice Representation , 1996 .

[154]  Philip Wadler,et al.  Comprehending monads , 1990, Mathematical Structures in Computer Science.

[155]  Alasdair Urquhart,et al.  Completeness of weak implication , 2008 .

[156]  Neil Tennant,et al.  The transmission of truth and the transitivity of deduction , 1994 .

[157]  Nick Benton,et al.  A Term Calculus for Intuitionistic Linear Logic , 1993, TLCA.

[158]  Robert K. Meyer,et al.  Gentzen’s Cut and Ackermann’s Gamma , 1989 .

[159]  N. Rescher.  Review: Moh Shaw-Kwei, The Deduction Theorems and Two New Logical Systems , 1952 .

[160]  David Gries,et al.  Programming Concepts and Methods , 1998 .

[161]  M. Black,et al.  Translations from the philosophical writings of Gottlob Frege , 1953 .

[162]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

[163]  J. Beall,et al.  Logical pluralism , 2000 .

[164]  Joachim Lambek,et al.  On the Calculus of Syntactic Types , 1961 .

[165]  Mariangiola Dezani-Ciancaglini,et al.  A filter lambda model and the completeness of type assignment , 1983, Journal of Symbolic Logic.

[166]  Richard B. White.  The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz , 1979, J. Philos. Log..

[167]  Greg Restall,et al.  Defining Double Negation Elimination , 2000, Log. J. IGPL.

[168]  J. Michael Dunn.  Algebraic Completeness Results for R-Mingle and Its Extensions , 1970, J. Symb. Log..

[169]  Haskell B. Curry,et al.  Combinatory Logic, Volume I , 1959 .

[170]  John K. Slaney,et al.  Reduced models for relevant logics without WI , 1987, Notre Dame J. Formal Log..

[171]  K. Jon Barwise,et al.  Imperfect information flow , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[172]  J. Cheney,et al.  A sequent calculus for nominal logic , 2004, LICS 2004.

[173]  Greg Restall,et al.  Information Flow and Relevant Logics , 1996 .

[174]  Dana,et al.  JSL volume 88 issue 4 Cover and Front matter , 1983, The Journal of Symbolic Logic.

[175]  P. Vos,et al.  http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless , 2007 .

[176]  Dave Barker-Plummer,et al.  Language, Proof and Logic , 1999 .

[177]  Hajnal Andréka,et al.  Decision problems for equational theories of relation algebras , 1997, Memoirs of the American Mathematical Society.

[178]  Kit Fine,et al.  Models for entailment , 1974, J. Philos. Log..

[179]  Graham Priest,et al.  Simplified semantics for basic relevant logics , 1992, J. Philos. Log..

[180]  Mati Pentus.  Free monoid completeness of the Lambek calculus allowing empty premises , 1998 .

[181]  Martin Odersky,et al.  Call-by-name, Call-by-value, Call-by-need and the Linear lambda Calculus , 1999, Theor. Comput. Sci..

[182]  Marcelo P Fiore,et al.  Topology via Logic , 1999 .

[183]  Nuel Belnap,et al.  Modalities in Ackermann's “rigorous implication” , 1959, Journal of Symbolic Logic.

[184]  Robert K. Meyer,et al.  On conserving positive logics , 1973, Notre Dame J. Formal Log..

[185]  G. Priest.  In Contradiction: A Study of the Transconsistent , 1987 .

[186]  Robert K. Meyer,et al.  ⊃E is Admissible in “true” relevant arithmetic , 1998, J. Philos. Log..

[187]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[188]  R. Routley,et al.  Relevantism, Material Detachment, and the Disjunctive Syllogism Argument , 1984, Canadian Journal of Philosophy.

[189]  Guttorm Fløistad.  Philosophy of language . Philosophical logic , 1981 .

[190]  Robert K. Meyer,et al.  Implementing the ‘Fool's model’ of combinatory logic , 1991, Journal of Automated Reasoning.

[191]  J. Roger Hindley,et al.  The simple semantics for Coppe-Dezani-Sallé types , 1982, Symposium on Programming.

[192]  Max Black,et al.  Translations from the philosophical writings of Gottlob Frege , 1953 .

[193]  D. Hubin,et al.  THE JOURNAL OF PHILOSOPHY , 2004 .

[194]  Nuel D. Belnap,et al.  A Decision Procedure For the System EĪ of Entailment with Negation , 1965 .

[195]  R. Meyer,et al.  Intuitionism, Entailment, Negation , 1973 .

[196]  Robert K. Meyer,et al.  E, R AND γ , 1969, Journal of Symbolic Logic.

[197]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[198]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[199]  Kosta Dosen,et al.  The first axiomatization of relevant logic , 1992, J. Philos. Log..

[200]  Nuel Belnap,et al.  Display logic , 1982, J. Philos. Log..

[201]  Alfred Tarski,et al.  Logic, Semantics, Metamathematics: Papers from 1923 to 1938 , 1958 .

[202]  J. Dunn,et al.  The algebra of intensional logics , 2019 .

[203]  Dov M. Gabbay,et al.  A general theory of the conditional in terms of a ternary operator , 2008 .

[204]  Dirk Roorda,et al.  Resource Logics : Proof-Theoretical Investigations , 1991 .

[205]  Ernest Lepore,et al.  Truth and interpretation : perspectives on the philosophy of Donald Davidson , 1989 .

[206]  Johan van Benthem,et al.  Language in action , 1991, J. Philos. Log..

[207]  R. Harrop,et al.  On disjunctions and existential statements in intuitionistic systems of logic , 1956 .

[208]  Greg Restall,et al.  Defending Logical Pluralism , 1999 .

[209]  Carl A. Gunter,et al.  Semantic Domains , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[210]  Dana S. Scott,et al.  Lambda Calculus: Some Models, Some Philosophy , 1980 .

[211]  Mati Pentus.  Models for the Lambek Calculus , 1995, Ann. Pure Appl. Log..

[212]  E. H. Hutten.  SEMANTICS , 1953, The British Journal for the Philosophy of Science.

[213]  J. M. Dunn,et al.  Truth or consequences , 1990 .

[214]  Philip Wadler,et al.  Is there a use for linear logic? , 1991, PEPM '91.

[215]  Stephen Read.  What is wrong with disjunctive syllogism , 1981 .

[216]  D. Scott.  Models for Various Type-Free Calculi , 1973 .

[217]  Ross T. Brady.  Relevant implication and the case for a weaker logic , 1996, J. Philos. Log..

[218]  Philip Wadler,et al.  Linear Types can Change the World! , 1990, Programming Concepts and Methods.

[219]  Chrysafis Hartonas.  Pretopology Semantics for Bimodal Intuitionistic Linear Logic , 1997, Log. J. IGPL.

[220]  Greg Restall,et al.  Simplified semantics for relevant logics (and some of their rivals) , 1993, J. Philos. Log..

[221]  Laurel Snyder,et al.  Sense , 1997, The Dramatic Text Workbook and Video.

[222]  Thomas Ehrhard.  Hypercoherences: A Strongly Stable Model of Linear Logic , 1993, Math. Struct. Comput. Sci..

[223]  Robert Goldblatt,et al.  Semantic analysis of orthologic , 1974, J. Philos. Log..

[224]  J. Roger Hindley,et al.  To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism , 1980 .

[225]  Michael A. McRobbie,et al.  Relevant analytic tableaux , 1979 .

[226]  Vincent Danos,et al.  The structure of multiplicatives , 1989, Arch. Math. Log..

[227]  Alasdair Urquhart,et al.  The undecidability of entailment and relevant implication , 1984, Journal of Symbolic Logic.

[228]  Valeria de Paiva,et al.  Poset-valued sets or how to build models for linear logics , 2004, Theor. Comput. Sci..

[229]  George Hutchinson,et al.  Recursively unsolvable word problems of modular lattices and diagram-chasing , 1973 .

[230]  Edwin D. Mares,et al.  On S , 1994, Stud Logica.

[231]  J. Michael Dunn,et al.  Positive modal logic , 1995, Stud Logica.

[232]  J. M. Dunn,et al.  Modern Uses of Multiple-Valued Logic , 1977 .

[233]  John D. Norton,et al.  The Cosmos of Science , 1997 .

[234]  Joachim Lambek.  Deductive systems and categories , 2005, Mathematical systems theory.

[235]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[236]  Greg Restall.  Four-valued semantics for relevant logics (and some of their rivals) , 1995, J. Philos. Log..

[237]  Giovanni Sambin,et al.  Topology and duality in modal logic , 1988, Ann. Pure Appl. Log..

[238]  A. Church.  The calculi of lambda-conversion , 1941 .

[239]  Richard Routley,et al.  An undecidable relevant logic , 1973 .

[240]  Lorenzo Peña y Gonzalo.  Paraconsistent logic: essays on the inconsistent , 1990 .

[241]  S. Weidenschilling,et al.  A plurality of worlds , 1991, Nature.

[242]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[243]  Robert K. Meyer,et al.  An Undecidability Result in the Theory of Relevant Implication , 1968 .

[244]  Wilhelm Ackermann,et al.  Begründung einer strengen Implikation , 1956, Journal of Symbolic Logic.

[245]  H. Wansing.  Substructural logics , 1996 .

[246]  Jean-Yves Girard.  Coherent Banach Spaces: A Continuous Denotational Semantics , 1999, Theor. Comput. Sci..

[247]  Greg Restall.  Laws of Non-Contradiction, Laws of the Excluded Middle, and Logics * , 2004 .

[248]  Nuel Belnap,et al.  The pure calculus of entailment , 1962, Journal of Symbolic Logic.

[249]  Nuel Belnap,et al.  A consecutive calculus for positive relevant implication with necessity , 1980, J. Philos. Log..

[250]  Heinrich Wansing,et al.  The Logic of Information Structures , 1993, Lecture Notes in Computer Science.

[251]  Nuel D. Belnap,et al.  Entailment and the disjunctive syllogism , 1981 .

[252]  Kit Fine,et al.  Semantics for quantified relevance logic , 1988, J. Philos. Log..

[253]  Paul C. Gilmore,et al.  The Consistency of Partial Set Theory without Extensionality , 1974 .

[254]  Greg Restall,et al.  Displaying and Deciding Substructural Logics 1: Logics with Contraposition , 1998, J. Philos. Log..

[255]  Kosta Dosen,et al.  A Brief Survey of Frames for the Lambek Calculus , 1992, Math. Log. Q..

[256]  Philip Wadler,et al.  There's No Substitute for Linear Logic , 1991 .

[257]  J. Dunn,et al.  Intuitive semantics for first-degree entailments and ‘coupled trees’ , 1976 .

[258]  Giovanni Sambin,et al.  Pretopologies and completeness proofs , 1995, Journal of Symbolic Logic.

[259]  Hiroakira Ono,et al.  Logics without the contraction rule , 1985, Journal of Symbolic Logic.

[260]  Richard Sylvan,et al.  The semantics of entailment — III , 1972, J. Philos. Log..

[261]  Ross T. Brady.  The simple consistency of a set theory based on the logic CSQ , 1983, Notre Dame J. Formal Log..

[262]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[263]  Giovanni Sambin.  Intuitionistic formal spaces and their neighbourhood , 1989 .