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  1. Samson Abramsky & Radha Jagadeesan (1994). Games and Full Completeness for Multiplicative Linear Logic. Journal of Symbolic Logic 59 (2):543-574.
    We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a (...)
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  2. V. Michele Abrusci (1991). Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic. Journal of Symbolic Logic 56 (4):1403-1451.
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  3. Romà J. Adillon & Ventura Verdú (2000). On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion. Studia Logica 65 (1):11-30.
    In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is a (...)
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  4. Natasha Alechina & Michiel van Lambalgen (1996). Generalized Quantification as Substructural Logic. Journal of Symbolic Logic 61 (3):1006-1044.
    We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...)
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  5. Majid Alizadeh, Farzaneh Derakhshan & Hiroakira Ono (forthcoming). Uniform Interpolation in Substructural Logics. Review of Symbolic Logic:1-30.
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  6. 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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  7. Roberto Arpaia (2004). On a Substructural Logic with Minimal Negation. Bulletin of the Section of Logic 33 (3):143-156.
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  8. C. M. Asmus (2009). Restricted Arrow. Journal of Philosophical Logic 38 (4):405 - 431.
    In this paper I present a range of substructural logics for a conditional connective ↦. This connective was original introduced semantically via restriction on the ternary accessibility relation R for a relevant conditional. I give sound and complete proof systems for a number of variations of this semantic definition. The completeness result in this paper proceeds by step-by-step improvements of models, rather than by the one-step canonical model method. This gradual technique allows for the additional control, lacking in the canonical (...)
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  9. 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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  10. Andrew Bacon (2013). A New Conditional for Naive Truth Theory. Notre Dame Journal of Formal Logic 54 (1):87-104.
    In this paper a logic for reasoning disquotationally about truth is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have heretofore failed to provide.
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  11. Andrew Bacon (2013). Curry's Paradox and Omega Inconsistency. Studia Logica 101 (1):1-9.
    In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...)
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  12. Andrew Bacon (2013). Paradoxes of Logical Equivalence and Identity. Topoi (1):1-10.
    In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.
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  13. Frank A. Bäuerle, David Albrecht, John N. Crossley & John S. Jeavons (1998). Curry-Howard Terms for Linear Logic. Studia Logica 61 (2):223-235.
    In this paper we 1. provide a natural deduction system for full first-order linear logic, 2. introduce Curry-Howard-style terms for this version of linear logic, 3. extend the notion of substitution of Curry-Howard terms for term variables, 4. define the reduction rules for the Curry-Howard terms and 5. outline a proof of the strong normalization for the full system of linear logic using a development of Girard's candidates for reducibility, thereby providing an alternative to Girard's proof using proof-nets.
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  14. Colin R. Caret & Zach Weber (2015). A Note on Contraction-Free Logic for Validity. Topoi 34 (1):63-74.
    This note motivates a logic for a theory that can express its own notion of logical consequence—a ‘syntactically closed’ theory of naive validity. The main issue for such a logic is Curry’s paradox, which is averted by the failure of contraction. The logic features two related, but different, implication connectives. A Hilbert system is proposed that is complete and non-trivial.
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  15. Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui (2012). Algebraic Proof Theory for Substructural Logics: Cut-Elimination and Completions. Annals of Pure and Applied Logic 163 (3):266-290.
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  16. Petr Cintula, Rostislav Horčík & Carles Noguera (2013). Nonassociative Substructural Logics and Their Semilinear Extensions: Axiomatization and Completeness Properties. Review of Symbolic Logic 6 (3):394-423.
    Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both (...)
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  17. Mikaël Cozic (2006). Epistemic Models, Logical Monotony and Substructural Logics. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 11--23.
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  18. Kosta Došen (1992). Modal Translations in Substructural Logics. Journal of Philosophical Logic 21 (3):283 - 336.
    Substructural logics are logics obtained from a sequent formulation of intuitionistic or classical logic by rejecting some structural rules. The substructural logics considered here are linear logic, relevant logic and BCK logic. It is proved that first-order variants of these logics with an intuitionistic negation can be embedded by modal translations into S4-type extensions of these logics with a classical, involutive, negation. Related embeddings via translations like the double-negation translation are also considered. Embeddings into analogues of S4 are obtained with (...)
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  19. Jacques Dubucs & Mathieu Marion (2003). Radical Anti-Realism and Substructural Logics. In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers. 235--249.
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  20. J. Michael Dunn, Mai Gehrke & Alessandra Palmigiano (2005). Canonical Extensions and Relational Completeness of Some Substructural Logics. Journal of Symbolic Logic 70 (3):713 - 740.
    In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.
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  21. Hadi Farahani & Hiroakira Ono (2012). Glivenko Theorems and Negative Translations in Substructural Predicate Logics. Archive for Mathematical Logic 51 (7-8):695-707.
    Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFL e . It is shown that there exists the weakest logic over QFL e among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are (...)
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  22. Josep Maria Font (2007). On Substructural Logics Preserving Degrees of Truth. Bulletin of the Section of Logic 36 (3/4):117-129.
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  23. Rohan French & David Ripley (forthcoming). Contractions of Noncontractive Consequence Relations. Review of Symbolic Logic.
    Some theorists have developed formal approaches to truth that depend on counterexamples to the structural rules of contraction. Here, we study such approaches, with an eye to helping them respond to a certain kind of objection. We define a contractive relative of each noncontractive relation, for use in responding to the objection in question, and we explore one example: the contractive relative of multiplicative-additive affine logic with transparent truth, or MAALT. -/- .
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  24. André Fuhrmann & Edwin D. Mares (1994). On S. Studia Logica 53 (1):75 - 91.
    The sentential logic S extends classical logic by an implication-like connective. The logic was first presented by Chellas as the smallest system modelled by contraining the Stalnaker-Lewis semantics for counterfactual conditionals such that the conditional is effectively evaluated as in the ternary relations semantics for relevant logics. The resulting logic occupies a key position among modal and substructural logics. We prove completeness results and study conditions for proceeding from one family of logics to another.
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  25. Ken-etsu Fujita (1998). On Proof Terms and Embeddings of Classical Substructural Logics. Studia Logica 61 (2):199-221.
    There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of the right structural rules. In terms (...)
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  26. Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier.
    This is also where we begin investigating lattices of logics and varieties, rather than particular examples.
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  27. Nikolaos Galatos & Hiroakira Ono (2006). Algebraization, Parametrized Local Deduction Theorem and Interpolation for Substructural Logics Over FL. Studia Logica 83 (1-3):279 - 308.
    Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
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  28. Nikolaos Galatos & Hiroakira Ono (2006). Glivenko Theorems for Substructural Logics Over FL. Journal of Symbolic Logic 71 (4):1353 - 1384.
    It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part (...)
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  29. Lou Goble (2007). Combinatory Logic and the Semantics of Substructural Logics. Studia Logica 85 (2):171 - 197.
    The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then (...)
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  30. Ai-ni Hsieh & James G. Raftery (2007). Conserving Involution in Residuated Structures. Mathematical Logic Quarterly 53 (6):583-609.
    This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which (...)
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  31. Lloyd Humberstone (2007). Investigations Into a Left-Structural Right-Substructural Sequent Calculus. Journal of Logic, Language and Information 16 (2):141-171.
    We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules (...)
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  32. R. Jones (2013). Robert Goldblatt. Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics. Lecture Notes in Logic; 38. Cambridge: Cambridge University Press, 2011. Isbn 978-1-107-01052-9. Pp. XIII + 282. [REVIEW] Philosophia Mathematica 21 (1):123-127.
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  33. N. Kamide (2002). A Canonical Model Construction For Substructural Logics With Strong Negation. Reports on Mathematical Logic:95-116.
    We introduce Kripke models for propositional substructural logics with strong negation, and show the completeness theorems for these logics using an extended Ishihara's canonical model construction method. The framework presented can deal with a broad range of substructural logics with strong negation, including a modified version of Nelson's logic N$^-$, Wansing's logic COSPL, and extended versions of Visser's basic propositional logic, positive relevant logics, Corsi's logics and M\'endez's logics.
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  34. Norihiro Kamide (2007). Synthesized Substructural Logics. Mathematical Logic Quarterly 53 (3):219-225.
    A mechanism for combining any two substructural logics (e.g. linear and intuitionistic logics) is studied from a proof-theoretic point of view. The main results presented are cut-elimination and simulation results for these combined logics called synthesized substructural logics.
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  35. Norihiro Kamide (2003). Normal Modal Substructural Logics with Strong Negation. Journal of Philosophical Logic 32 (6):589-612.
    We introduce modal propositional substructural logics with strong negation, and prove the completeness theorems (with respect to Kripke models) for these logics.
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  36. Norihiro Kamide (2002). Kripke Semantics for Modal Substructural Logics. Journal of Logic, Language and Information 11 (4):453-470.
    We introduce Kripke semantics for modal substructural logics, and provethe completeness theorems with respect to the semantics. Thecompleteness theorems are proved using an extended Ishihara's method ofcanonical model construction (Ishihara, 2000). The framework presentedcan deal with a broad range of modal substructural logics, including afragment of modal intuitionistic linear logic, and modal versions ofCorsi's logics, Visser's logic, Méndez's logics and relevant logics.
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  37. Norihiro Kamide (2002). Substructural Logics with Mingle. Journal of Logic, Language and Information 11 (2):227-249.
    We introduce structural rules mingle, and investigatetheorem-equivalence, cut- eliminability, decidability, interpolabilityand variable sharing property for sequent calculi having the mingle.These results include new cut-elimination results for the extendedlogics: FLm (full Lambek logic with the mingle), GLm(Girard's linear logic with the mingle) and Lm (Lambek calculuswith restricted mingle).
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  38. Sato Kentaro (2008). Proper Semantics for Substructural Logics, From a Stalker Theoretic Point of View. Studia Logica 88 (2):295 - 324.
    We study filters in residuated structures that are associated with congruence relations (which we call -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters. We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of -filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters will turn out to coincide with truth sets under various (...)
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  39. Kentaro Kikuchi (2001). Relationships Between Basic Propositional Calculus and Substructural Logics. Bulletin of the Section of Logic 30 (1):15-20.
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  40. Nils Kürbis (2008). Stable Harmony. In Peliš Michal (ed.), Logica Yearbook 2007.
    In this paper, I'll present a general way of "reading off" introduction/elimination rules from elimination/introduction rules, and define notions of harmony and stability on the basis of it.
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  41. Ondrej Majer & Michal Pelis (2013). Knowledge Interpretation in Substructural Frames. Organon F 20:79-98.
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  42. Edwin Mares & Francesco Paoli (2014). Logical Consequence and the Paradoxes. Journal of Philosophical Logic 43 (2-3):439-469.
    We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, (...)
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  43. George Metcalfe (2009). A Sequent Calculus for Constructive Logic with Strong Negation as a Substructural Logic. Bulletin of the Section of Logic 38 (1):1-7.
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  44. Motohiko Mouri & Norihiro Kamide (2008). Strong Normalizability of Typed Lambda-Calculi for Substructural Logics. Logica Universalis 2 (2):189-207.
    The strong normalization theorem is uniformly proved for typed λ-calculi for a wide range of substructural logics with or without strong negation.
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  45. H. Naruse, Bayu Surarso & H. Ono (1998). A Syntactic Approach to Maksimova's Principle of Variable Separation for Some Substructural Logics. Notre Dame Journal of Formal Logic 39 (1):94-113.
    Maksimova's principle of variable separation says that if propositional formulas $A_1 \supset A_2$ and $B_1 \supset B_2$ have no propositional variables in common and if a formula $A_1\wedge B_1 \supset A_2\vee B_2$ is provable, then either $A_1 \supset A_2$ or $B_1 \supset B_2$ is provable. Results on Maksimova's principle until now are obtained mostly by using semantical arguments. In the present paper, a proof-theoretic approach to this principle in some substructural logics is given, which analyzes a given cut-free proof of (...)
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  46. Francesco Paoli (2005). The Ambiguity of Quantifiers. Philosophical Studies 124 (3):313 - 330.
    In the tradition of substructural logics, it has been claimed for a long time that conjunction and inclusive disjunction are ambiguous:we should, in fact, distinguish between ‘lattice’ connectives (also called additive or extensional) and ‘group’ connectives (also called multiplicative or intensional). We argue that an analogous ambiguity affects the quantifiers. Moreover, we show how such a perspective could yield solutions for two well-known logical puzzles: McGee’s counterexample to modus ponens and the lottery paradox.
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  47. Greg Restall, Relevant and Substructural Logics.
    This essay is structured around the bifurcation between proofs and models: The first section discusses Proof Theory of relevant and substructural logics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, (...)
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  48. Greg Restall (forthcoming). Substructural Logics. Stanford Encyclopedia of Philosophy.
    summary of work in relevant in the Anderson– tradition.]; Mares Troestra, Anne, 1992, Lectures on , CSLI Publications [A quick, easy-to.
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  49. Greg Restall (2000). An Introduction to Substructural Logics. Routledge.
    This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both (...)
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  50. Greg Restall (1998). Displaying and Deciding Substructural Logics 1: Logics with Contraposition. [REVIEW] Journal of Philosophical Logic 27 (2):179-216.
    Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic (Belnap, 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics.
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