Results for 'substructural'

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  1. Comparing Substructural Theories of Truth.David Ripley - 2015 - Ergo: An Open Access Journal of Philosophy 2.
    Substructural theories of truth are theories based on logics that do not include the full complement of usual structural rules. Existing substructural approaches fall into two main families: noncontractive approaches and nontransitive approaches. This paper provides a sketch of these families, and argues for two claims: first, that substructural theories are better-positioned than other theories to grapple with the truth-theoretic paradoxes, and second—more tentatively—that nontransitive approaches are in turn better-positioned than noncontractive approaches.
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  2.  24
    Substructural Logics: A Primer.Francesco Paoli - 2002 - Dordrecht, Netherland: Springer.
    The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.
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  3.  70
    Substructural epistemic logics.Igor Sedlár - 2015 - Journal of Applied Non-Classical Logics 25 (3):256-285.
    The article introduces substructural epistemic logics of belief supported by evidence. The logics combine normal modal epistemic logics with distributive substructural logics. Pieces of evidence are represented by points in substructural models and availability of evidence is modelled by a function on the point set. The main technical result is a general completeness theorem. Axiomatisations are provided by means of two-sorted Hilbert-style calculi. It is also shown that the framework presents a natural solution to the problem of (...)
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  4.  88
    Craig interpolation for semilinear substructural logics.Enrico Marchioni & George Metcalfe - 2012 - Mathematical Logic Quarterly 58 (6):468-481.
    The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R-mingle with unit” logic that have the Craig interpolation property. This latter characterization is obtained using a model-theoretic quantifier elimination strategy to determine the varieties of Sugihara (...)
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  5.  92
    Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.
    Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these (...)
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  6.  34
    A Substructural Gentzen Calculus for Orthomodular Quantum Logic.Davide Fazio, Antonio Ledda, Francesco Paoli & Gavin St John - 2023 - Review of Symbolic Logic 16 (4):1177-1198.
    We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one (...)
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  7. Substructural logics, pluralism and collapse.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2018 - Synthese 198 (Suppl 20):4991-5007.
    When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a (...)
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  8.  39
    Substructural Logics.Peter Joseph Schroeder-Heister & Kosta Došen - 1993 - Oxford, England: Oxford University Press on Demand.
    The new area of logic and computation is now undergoing rapid development. This has affected the social pattern of research in the area. A new topic may rise very quickly with a significant body of research around it. The community, however, cannot wait the traditional two years for a book to appear. This has given greater importance to thematic collections of papers, centred around a topic and addressing it from several points of view, usually as a result of a workshop, (...)
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  9. Nonassociative substructural logics and their semilinear extensions: Axiomatization and completeness properties: Nonassociative substructural logics.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, (...)
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  10.  58
    Substructural logics, pragmatic enrichment, and the inferential role of logical constants.Pilar Terrés Villalonga - 2020 - Inquiry: An Interdisciplinary Journal of Philosophy 63 (6):628-654.
    ABSTRACT My aim in this paper is to present a pluralist thesis about the inferential role of logical constants, which embraces classical, relevant, linear and ordered logic. That is, I defend that a logical constant c has more than one correct inferential role. The thesis depends on a particular interpretation of substructural logics' vocabulary, according to which classical logic captures the literal meaning of logical constants and substructural logics encode a pragmatically enriched sense of those connectives. The paper (...)
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  11.  47
    Substructural heresies.Bogdan Dicher - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    The past decades have seen remarkable progress in the study of substructural logics, be it mathematically or philosophically oriented. This progress has a somewhat perplexing effect: the more subst...
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    Substructure lattices and almost minimal end extensions of models of Peano arithmetic.James H. Schmerl - 2004 - Mathematical Logic Quarterly 50 (6):533-539.
    This paper concerns intermediate structure lattices Lt, where [MATHEMATICAL SCRIPT CAPITAL N] is an almost minimal elementary end extension of the model ℳ of Peano Arithmetic. For the purposes of this abstract only, let us say that ℳ attains L if L ≅ Lt for some almost minimal elementary end extension of [MATHEMATICAL SCRIPT CAPITAL N]. If T is a completion of PA and L is a finite lattice, then: If some model of T attains L, then every countable model (...)
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  13.  72
    Substructural approaches to paradox: an introduction to the special issue.Elia Zardini - 2021 - Synthese 199 (3):493-525.
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  14.  25
    A Substructural Approach to Explicit Modal Logic.Shawn Standefer - 2023 - Journal of Logic, Language and Information 32 (2):333–362.
    In this paper, we build on earlier work by Standefer (Logic J IGPL 27(4):543–569, 2019) in investigating extensions of substructural logics, particularly relevant logics, with the machinery of justification logics. We strengthen a negative result from the earlier work showing a limitation with the canonical model method of proving completeness. We then show how to enrich the language with an additional operator for implicit commitment to circumvent these problems. We then extend the logics with axioms for D, 4, and (...)
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  15.  19
    Some observations on the substructure lattice of a 1 ultrapower.Thomas G. McLaughlin - 2010 - Mathematical Logic Quarterly 56 (3):323-330.
    Given a Δ1 ultrapower ℱ/[MATHEMATICAL SCRIPT CAPITAL U], let ℒU denote the set of all Π2-correct substructures of ℱ/[MATHEMATICAL SCRIPT CAPITAL U]; i.e., ℒU is the collection of all those subsets of |ℱ/[MATHEMATICAL SCRIPT CAPITAL U]| that are closed under computable functions. Defining in the obvious way the lattice ℒ) with domain ℒU, we obtain some preliminary results about lattice embeddings into – or realization as – an ℒ. The basis for these results, as far as we take the matter, (...)
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  16.  26
    Substructural Nuclear (Image-Based) Logics and Operational Kripke-Style Semantics.Eunsuk Yang - 2024 - Studia Logica 112 (4):805-833.
    This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective _N_ satisfying nucleus property, called here substructural _nuclear_ logics, and its subclass, called here substructural _nuclear image-based_ logics, where _N_ further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce _operational Kripke-style_ semantics for those logics and provide two sorts (...)
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    Substructural Negations.Takuro Onishi - 2015 - Australasian Journal of Logic 12 (4):177-203.
    We present substructural negations, a family of negations classified in terms of structural rules of an extended kind of sequent calculus, display calculus. In considering the whole picture, we emphasize the duality of negation. Two types of negative modality, impossibility and unnecessity, are discussed and "self-dual" negations like Classical, De Morgan, or Ockham negation are redefined as the fusions of two negative modalities. We also consider how to identify, using intuitionistic and dual intuitionistic negations, two accessibility relations associated with (...)
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  18.  68
    Informational interpretation of substructural propositional logics.Heinrich Wansing - 1993 - Journal of Logic, Language and Information 2 (4):285-308.
    This paper deals with various substructural propositional logics, in particular with substructural subsystems of Nelson's constructive propositional logics N– and N. Doen's groupoid semantics is extended to these constructive systems and is provided with an informational interpretation in terms of information pieces and operations on information pieces.
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  19.  11
    Substructural Negations as Normal Modal Operators.Heinrich Wansing - 2024 - In Yale Weiss & Romina Birman (eds.), Saul Kripke on Modal Logic. Cham: Springer. pp. 365-388.
    A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a (...)
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  20. Collection Frames for Distributive Substructural Logics.Greg Restall & Shawn Standefer - 2023 - Review of Symbolic Logic 16 (4):1120-1157.
    We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete (...)
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  21.  97
    An Introduction to Substructural Logics.Greg Restall - 1999 - New York: 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 (...)
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  22.  25
    Synthesized substructural logics.Norihiro Kamide - 2007 - 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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  23.  68
    Glivenko Theorems for Substructural Logics over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - 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 (...)
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  24.  50
    Substructural inquisitive logics.Vít Punčochář - 2019 - Review of Symbolic Logic 12 (2):296-330.
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  25.  95
    Substructural Fuzzy-Relevance Logic.Eunsuk Yang - 2015 - Notre Dame Journal of Formal Logic 56 (3):471-491.
    This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication and its neighbors. The algebraic structures corresponding (...)
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    Substructures of a nation according to cardinal Stefan wyszynski (substruktury narodu wedlug kardynala Stefana wyszynskiego).Ziabski Grzegorz - 2009 - Archeus. Studia Z Bioetyki I Antropologii Filozoficznej 10:123-136.
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    Small substructures and decidability issues for first-order logic with two variables.Emanuel Kieroński & Martin Otto - 2012 - Journal of Symbolic Logic 77 (3):729-765.
    We study first-order logic with two variables FO² and establish a small substructure property. Similar to the small model property for FO² we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO² under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO² has the finite model property and is complete for non-deterministic (...)
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    Connexive Implications in Substructural Logics.Davide Fazio & Gavin St John - 2024 - Review of Symbolic Logic 17 (3):878-909.
    This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, (...)
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  29. Kripke semantics for modal substructural logics.Norihiro Kamide - 2002 - 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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  30.  68
    Normal modal substructural logics with strong negation.Norihiro Kamide - 2003 - 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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  31. 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 (...)
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  32.  15
    Distributive Substructural Logics as Coalgebraic Logics over Posets.Marta Bílková, Rostislav Horčik & Jiří Velebil - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 119-142.
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    Metainferentially substructural validity theories.Federico Pailos - forthcoming - Journal of Applied Non-Classical Logics:1-22.
    1. As Graham Priest claims in Priest (forthcoming), blocking semantic paradoxes is not hard. It just requires giving up (at least) one of the principles involved in the derivation of the undesirabl...
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    Structural Completeness in Substructural Logics.J. S. Olson, J. G. Raftery & C. J. Van Alten - 2008 - Logic Journal of the IGPL 16 (5):453-495.
    Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.
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  35.  73
    Metacompleteness of Substructural Logics.Takahiro Seki - 2012 - Studia Logica 100 (6):1175-1199.
    Metacompleteness is used to prove properties such as the disjunction property and the existence property in the area of relevant logics. On the other hand, the disjunction property of several basic propositional substructural logics over FL has been proved using the cut elimination theorem of sequent calculi and algebraic characterization. The present paper shows that Meyer’s metavaluational technique and Slaney’s metavaluational technique can be applied to basic predicate intuitionistic substructural logics and basic predicate involutive substructural logics, respectively. (...)
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    Higher-Level Paradoxes and Substructural Solutions.Rashed Ahmad - forthcoming - Studia Logica:1-25.
    There have been recent arguments against the idea that substructural solutions are uniform. The claim is that even if the substructuralist solves the common semantic paradoxes uniformly by targeting Cut or Contraction, with additional machinery, we can construct higher-level paradoxes (e.g., a higher-level Liar, a higher-level Curry, and a meta-validity Curry). These higher-level paradoxes do not use metainferential Cut or Contraction, but rather, higher-level Cuts and higher-level Contractions. These kinds of paradoxes suggest that targeting Cut or Contraction is not (...)
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  37. Substructural Logics.Peter Schroeder-Heister - 1996 - Erkenntnis 45 (1):115-118.
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  38.  35
    Algebraic Perspectives on Substructural Logics.Davide Fazio, Antonio Ledda & Francesco Paoli (eds.) - 2020 - Springer International Publishing.
    This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. -/- Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have (...)
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    Variable Sharing in Substructural Logics: An Algebraic Characterization.Guillermo Badia - 2018 - Bulletin of the Section of Logic 47 (2):107-115.
    We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties. -/- .
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  40.  40
    An Outline of a Substructural Model of BTA Belief.Igor Sedlar - 2013 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 20 (2):160-170.
    The paper outlines an epistemic logic based on the proof theory of substructural logics. The logic is a formal model of belief that i) is based on true assumptions (BTA belief) and ii) does not suffer from the usual omniscience properties.
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  41.  41
    Editorial Introduction: Substructural Logics and Metainferences.Eduardo Barrio & Paul Égré - 2022 - Journal of Philosophical Logic 51 (6):1215-1231.
    The concept of _substructural logic_ was originally introduced in relation to limitations of Gentzen’s structural rules of Contraction, Weakening and Exchange. Recent years have witnessed the development of substructural logics also challenging the Tarskian properties of Reflexivity and Transitivity of logical consequence. In this introduction we explain this recent development and two aspects in which it leads to a reassessment of the bounds of classical logic. On the one hand, standard ways of defining the notion of logical consequence in (...)
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  42.  6
    Metainferentially substructural validity theories.Germany Tübingen - forthcoming - Journal of Applied Non-Classical Logics:1-22.
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  43.  49
    Canonicity results of substructural and lattice-based logics.Tomoyuki Suzuki - 2011 - Review of Symbolic Logic 4 (1):1-42.
    In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach (...)
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  44.  71
    The Substructure of stasis-theory from Hermagoras to Hermogenes.Malcolm Heath - 1994 - Classical Quarterly 44 (01):114-.
    Stasis-theory seeks to classify rhetorical problems acccording to the underlying structure of the dispute that each involves. Such a classification is of interest to the practising rhetor, since it may help him identify an appropriate argumentative strategy; for example, patterns of argument appropriate to a question of fact may be irrelevant in an evaluative dispute.
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  45. Relevant and substructural logics.Greg Restall - unknown
    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 (...)
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  46.  14
    (1 other version)Models for Substructural Arithmetics.Greg Restall - 2010 - Australasian Journal of Logic 8:82-99.
    This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R.
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  47.  43
    Strong Normalizability of Typed Lambda-Calculi for Substructural Logics.Motohiko Mouri & Norihiro Kamide - 2008 - 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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  48.  23
    Reply to ‘On substructure in titanium alloy martensite’.Tomonari Inamura, Hideki Hosoda, Hee Young Kim & Shuichi Miyazaki - 2011 - Philosophical Magazine 91 (16):2079-2080.
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    An Algebraic Approach to the Disjunction Property of Substructural Logics.Daisuke Souma - 2007 - Notre Dame Journal of Formal Logic 48 (4):489-495.
    Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of (...)
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  50. Displaying and deciding substructural logics 1: Logics with contraposition.Greg Restall - 1998 - 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. 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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