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  1. Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic.V. Michele Abrusci - 1991 - Journal of Symbolic Logic 56 (4):1403-1451.
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  2. On a Substructural Gentzen System, its Equivalent Variety Semantics and its External Deductive System.R. Adillon & Ventura Verdú - 2002 - Bulletin of the Section of Logic 31 (3):125-134.
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  3. On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion.Romà J. Adillon & Ventura Verdú - 2000 - 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. Generalized Quantification as Substructural Logic.Natasha Alechina & Michiel van Lambalgen - 1996 - 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. Generalized Quantification as Substructural Logic.Natasha Alechina & Michiel van Lambalgen - 1996 - 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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  6. Uniform Interpolation in Substructural Logics.Majid Alizadeh, Farzaneh Derakhshan & Hiroakira Ono - 2014 - Review of Symbolic Logic 7 (3):455-483.
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  7. Noisy Vs. Merely Equivocal Logics.Patrick Allo - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 57--79.
    Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these connectives cannot simultaneously exhibit all these properties. This is just a more sophisticated way of saying that while natural language connectives are ambiguous, they are not so in the way classical logic intends them to be. Since this view is usually framed as a means to resolve (...)
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  8. Information and Logical Discrimination.Patrick Allo - 2012 - In S. Barry Cooper (ed.), How the World Computes. pp. 17--28.
    Informational conceptions of logic are barely novel. We find them in the work of John Corcoran, in several papers on substructural and constructive logics by Heinrich Wansing, and in the interpretation of the Routley-Meyer semantics for relevant logics in terms of Barwises and Perrys theory of situations. Allo & Mares [2] present an informational account of logical consequence that is based on the content-nonexpantion platitude, but that also relies on a double inversion of the standard direction of explanation (in- formation (...)
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  9. Local Information and Adaptive Consequence.Patrick Allo - 2006 - Logique Et Analyse 149:461-488.
    In this paper we provide a formal description of what it means to be in a local or partial information-state. Starting from the notion of locality in a relational structure, we define so-called adaptive gen- erated submodels. The latter are then shown to yield an adaptive logic wherein the derivability of Pφ is naturally interpreted as a core property of being in a state in which one holds the information that φ.
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  10. 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, (...)
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  11. On a Substructural Logic with Minimal Negation.Roberto Arpaia - 2004 - Bulletin of the Section of Logic 33 (3):143-156.
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  12. Restricted Arrow.C. M. Asmus - 2009 - 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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  13. 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, (...)
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  14. The Semantics and Proof Theory of Linear Logic.Arnon Avron - 1988 - Theoretical Computer Science 57:161-184.
    Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we shall investigate (...)
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  15. Paradoxes of Logical Equivalence and Identity.Andrew Bacon - 2013 - 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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  16. A New Conditional for Naive Truth Theory.Andrew Bacon - 2013 - 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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  17. Curry's Paradox and Omega Inconsistency.Andrew Bacon - 2013 - 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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  18. 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.
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  19. 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.
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  20. The Logics of Strict-Tolerant Logic.Eduardo Barrio, Lucas Rosenblatt & Diego Tajer - 2015 - Journal of Philosophical Logic 44 (5):551-571.
    Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egré, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this (...)
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  21. Curry-Howard Terms for Linear Logic.Frank A. Bäuerle, David Albrecht, John N. Crossley & John S. Jeavons - 1998 - 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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  22. Current Trends in Substructural Logics.Katalin Bimbó - forthcoming - Journal of Philosophical Logic:1-16.
    This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic.
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  23. Investigation Into Combinatory Systems with Dual Combinators.Katalin Bimbó - 2000 - Studia Logica 66 (2):285-296.
    Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications, led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators, the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser (...)
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  24. Substructural Logics, Combinatory Logic, and Lambda-Calculus.Katalin Bimbo - 1999 - Dissertation, Indiana University
    The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker (...)
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  25. Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi.Katalin Bimbo & J. Michael Dunn - 2008 - Center for the Study of Language and Inf.
    Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.
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  26. Sequential Dynamic Logic.Alexander Bochman & Dov M. Gabbay - 2012 - Journal of Logic, Language and Information 21 (3):279-298.
    We introduce a substructural propositional calculus of Sequential Dynamic Logic that subsumes a propositional part of dynamic predicate logic, and is shown to be expressively equivalent to propositional dynamic logic. Completeness of the calculus with respect to the intended relational semantics is established.
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  27. Adding Logic to the Toolbox of Molecular Biology.Giovanni Boniolo, Marcello D’Agostino, Mario Piazza & Gabriele Pulcini - 2015 - European Journal for Philosophy of Science 5 (3):399-417.
    The aim of this paper is to argue that logic can play an important role in the “toolbox” of molecular biology. We show how biochemical pathways, i.e., transitions from a molecular aggregate to another molecular aggregate, can be viewed as deductive processes. In particular, our logical approach to molecular biology — developed in the form of a natural deduction system — is centered on the notion of Curry-Howard isomorphism, a cornerstone in nineteenth-century proof-theory.
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  28. Which Logic for the Radical Anti-Realist ?Denis Bonnay & Mikaël Cozic - unknown -
    Since the ground-breaking contributions of M. Dummett (Dummett 1978), it is widely recognized that anti-realist principles have a critical impact on the choice of logic. Dummett argued that classical logic does not satisfy the requirements of such principles but that intuitionistic logic does. Some philosophers have adopted a more radical stance and argued for a more important departure from classical logic on the basis of similar intuitions. In particular, J. Dubucs and M. Marion (?) and (Dubucs 2002) have recently argued (...)
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  29. On Two Fragments with Negation and Without Implication of the Logic of Residuated Lattices.Félix Bou, Àngel García-Cerdaña & Ventura Verdú - 2006 - Archive for Mathematical Logic 45 (5):615-647.
    The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], H BCK [36] (nowadays called by Ono), etc. In this paper we study the -fragment and the -fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It results that this (...)
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  30. The Low Stress Creep of Aluminium Near to the Melting Point: The Influence of Oxidation and Substructural Changes.B. Burton - 1972 - Philosophical Magazine 25 (3):645-659.
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  31. Finite Models of Some Substructural Logics.Wojciech Buszkowski - 2002 - Mathematical Logic Quarterly 48 (1):63-72.
    We give a proof of the finite model property of some fragments of commutative and noncommutative linear logic: the Lambek calculus, BCI, BCK and their enrichments, MALL and Cyclic MALL. We essentially simplify the method used in [4] for proving fmp of BCI and the Lambek ca culus and in [5] for proving fmp of MALL. Our construction of finite models also differs from that used in Lafont [8] in his proof of fmp of MALL.
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  32. Logical Consequence: Its Nature, Structure, and Application.Colin R. Caret & Ole T. Hjortland - 2015 - In Colin R. Caret & Ole T. Hjortland (eds.), Foundations of Logical Consequence. Oxford University Press.
    Recent work in philosophical logic has taken interesting and unexpected turns. It has seen not only a proliferation of logical systems, but new applications of a wide range of different formal theories to philosophical questions. As a result, philosophers have been forced to revisit the nature and foundation of core logical concepts, chief amongst which is the concept of logical consequence. This essay sets the contributions of the volume in context and identifies how they advance important debates within the philosophy (...)
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  33. A Note on Contraction-Free Logic for Validity.Colin R. Caret & Zach Weber - 2015 - 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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  34. Substructural Development During Strain Cycling of Alpha-Iron.O. K. Chopra & C. V. B. Gowda - 1974 - Philosophical Magazine 30 (3):583-591.
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  35. Note on Deduction Theorems in Contraction‐Free Logics.Karel Chvalovský & Petr Cintula - 2012 - Mathematical Logic Quarterly 58 (3):236-243.
    This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics . An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.
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  36. Algebraic Proof Theory for Substructural Logics: Cut-Elimination and Completions.Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui - 2012 - Annals of Pure and Applied Logic 163 (3):266-290.
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  37. Nonassociative Substructural Logics and Their Semilinear Extensions: Axiomatization and Completeness Properties.Petr Cintula, Rostislav Horčík & Carles Noguera - 2013 - 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 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 substructural logics, a form of (...)
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  38. Is Multiset Consequence Trivial?Petr Cintula & Francesco Paoli - forthcoming - Synthese:1-25.
    Dave Ripley has recently argued against the plausibility of multiset consequence relations and of contraction-free approaches to paradox. For Ripley, who endorses a nontransitive theory, the best arguments that buttress transitivity also push for contraction—whence it is wiser for the substructural logician to go nontransitive from the start. One of Ripley’s allegations is especially insidious, since it assumes the form of a trivialisation result: it is shown that if a multiset consequence relation can be associated to a closure operator in (...)
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  39. Epistemic Models, Logical Monotony and Substructural Logics.Mikaël Cozic - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 11--23.
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  40. Grafting Modalities Onto Substructural Implication Systems.Marcello D'agostino, Dov M. Gabbay & Alessandra Russo - 1997 - Studia Logica 59 (1):65-102.
    We investigate the semantics of the logical systems obtained by introducing the modalities and into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in (...)
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  41. An Introduction to Substructural Logics.Kosta Došen - 2001 - Bulletin of Symbolic Logic 7 (4):527-530.
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  42. Restall Greg. An Introduction to Substructural Logics. Routledge, London and New York 2000, Xiv+ 381 Pp. [REVIEW]Kosta Došen - 2001 - Bulletin of Symbolic Logic 7 (4):527-530.
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  43. Review: Greg Restall, An Introduction to Substructural Logics. [REVIEW]Kosta Dosen - 2001 - Bulletin of Symbolic Logic 7 (4):527-530.
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  44. Modal Logic as Metalogic.Kosta Došen - 1992 - Journal of Logic, Language and Information 1 (3):173-201.
    The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one (...)
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  45. Modal Translations in Substructural Logics.Kosta Došen - 1992 - 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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  46. Radical Anti-Realism and Substructural Logics.Jacques Dubucs & Mathieu Marion - 2003 - In A. Rojszczak, J. Cachro & G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Kluwer Academic Publishers. pp. 235--249.
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  47. Canonical Extensions and Relational Completeness of Some Substructural Logics.J. Michael Dunn, Mai Gehrke & Alessandra Palmigiano - 2005 - 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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  48. Combinators and Structurally Free Logic.J. Dunn & R. Meyer - 1997 - Logic Journal of the IGPL 5 (4):505-537.
    A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving a representation theorem for 'combinatory posets.' A philosophical interpretation is given of the models, showing that an element of a combinatory poset can be understood simultaneously as a set of states and as a set of actions on states. This double interpretation allows for (...)
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  49. Cut Elimination for a Calculus with Context-Dependent Rules.Birgit Elbl - 2001 - Archive for Mathematical Logic 40 (3):167-188.
    Context-dependent rules are an obstacle to cut elimination. Turning to a generalised sequent style formulation using deep inferences is helpful, and for the calculus presented here it is essential. Cut elimination is shown for a substructural, multiplicative, pure propositional calculus. Moreover we consider the extra problems induced by non-logical axioms and extend the results to additive connectives and quantifiers.
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  50. Glivenko Theorems and Negative Translations in Substructural Predicate Logics.Hadi Farahani & Hiroakira Ono - 2012 - 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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