About this topic
Summary Substructural logics are generally (though not universally) studied within a specific style of proof theory: sequent calculus. The reason is that sequent proofs draw a clear distinction between operational and structural rules. A substructural logic lacks some of the standards structural rules. Whereas each operational rule governs certain 'moves' involving a specific logical operator, the structural rules just change the arrangement of formulas 'around' the sign for implication or consequence (the sequent separator). Thus, we can think of structural rules as encoding meta-logical properties of the relation of logical consequence in the abstract, without reference to any specific operator of the object-language. The standard structural rules roughly correspond to properties such as reflexivity, contraction, and transitivity of the consequence relation. Substructural logics encode conceptions of consequence that lack some of these standard properties.
Key works The distinction between operational and structural rules comes from Gentzen 1964. For a textbook treatment of substructural logics, see Restall 1999 or Paoli 2002. The philosophical interest in these formal systems was driven, in one way, by connections with issues in information theory and computer science (Girard 1995) and, in another way, by connections to the tradition of relevant logic (Slaney 1990). More recently, there has been a trend of treating philosophical paradoxes in some fashion through formalization in substructural logic. For example, Cobreros et al 2012 apply this strategy to vagueness and the sorites, while Zardini 2011 and Ripley 2012 focus on the Liar paradox and related phenomena. 
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  1. 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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  2. Cut elimination for systems of transparent truth with restricted initial sequents.Carlo Nicolai - manuscript
    The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we (...)
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  3. Systems for non-reflexive consequence.Carlo Nicolai & Lorenzo Rossi - manuscript
    Substructural logics and their application to logical and semantic paradoxes have been extensively studied, but non-reflexive systems have been somewhat neglected. Here, we aim to fill this lacuna, at least in part, by presenting a non-reflexive logic and theory of naive consequence (and truth). We also investigate the semantics and the proof-theory of the system. Finally, we develop a compositional theory of truth (and consequence) in our non-reflexive framework.
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  4. Involutive Commutative Residuated Lattice without Unit: Logics and Decidability.Yiheng Wang, Hao Zhan, Yu Peng & Zhe Lin - manuscript
    We investigate involutive commutative residuated lattices without unit, which are commutative residuated lattice-ordered semigroups enriched with a unary involutive negation operator. The logic of this structure is discussed and the Genzten-style sequent calculus of it is presented. Moreover, we prove the decidability of this logic.
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  5. Non-reflexive Nonsense: Proof-Theory for Paracomplete Weak Kleene Logic.Bruno da Re, Damian Szmuc & M. Ines Corbalan - forthcoming - Studia Logica.
    Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic `of nonsense' introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic K3W by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) (...)
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  6. Connexive Implications in Substructural Logics.Davide Fazio & Gavin St John - forthcoming - Review of Symbolic Logic:1-32.
    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, we (...)
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  7. Semantics for Second Order Relevant Logics.Shay Logan - forthcoming - In Andrew Tedder, Shawn Standefer & Igor Sedlár (eds.), New Directions in Relevant Logic. Springer. pp. 211-226.
    Here's the thing: when you look at it from just the right angle, it's entirely obvious how semantics for second-order relevant logics ought to go. Or at least, if you've understood how semantics for first-order relevant logics ought to go, there are perspectives like this. What's more is that from any such angle, the metatheory that needs doing can be summed up in one line: everything is just as in the first-order case, but with more indices. Of course, it's no (...)
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  8. Frege meets Belnap: Basic Law V in a Relevant Logic.Shay Logan & Francesca Boccuni - forthcoming - In Andrew Tedder, Shawn Standefer & Igor Sedlar (eds.), New Directions in Relevant Logic. Springer. pp. 381-404.
    Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, (...)
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  9. Substructural Logics.Greg Restall - forthcoming - 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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  10. Dynamic consequence for soft information.Olivier Roy & Ole Thomassen Hjortland - forthcoming - Journal of Logic and Computation.
  11. Variable-Sharing as Relevance.Shawn Standefer - forthcoming - In Igor Sedlár, Shawn Standefer & Andrew Tedder (eds.), New Directions in Relevant Logic.
  12. Dialetheism and distributed sorites.Ben Blumson - 2023 - Synthese 202 (4):1-18.
    Noniterative approaches to the sorites paradox accept single steps of soritical reasoning, but deny that these can be combined into valid chains of soritical reasoning. The distributed sorites is a puzzle designed to undermine noniterative approaches to the sorites paradox, by deriving an inconsistent conclusion using only single steps, but not chains, of soritical reasoning. This paper shows how a dialetheist version of the noniterative approach, the strict-tolerant approach, also solves the distributed sorites paradox, at no further cost, by accepting (...)
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  13. ‘Forms of Conditionality’: Response to ‘Truth-Maker Semantics for Some Substructural Logics’ by Ondrej Majer, Vít Punčochář and Igor Sedlár.Kit Fine - 2023 - In Federico L. G. Faroldi & Frederik Van De Putte (eds.), Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Springer Verlag. pp. 223-230.
    In the light of the contribution of Majer, Punčochář and Sedlár, I develop a path-theoretic semantics for the conditional and briefly consider some of its applications.
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  14. Truth-Maker Semantics for Some Substructural Logics.Ondrej Majer, Vít Punčochář & Igor Sedlár - 2023 - In Federico L. G. Faroldi & Frederik Van De Putte (eds.), Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Springer Verlag. pp. 207-222.
    Fine (J Philos Log 43:549–577, 2014) developed a truthmaker semantics for intuitionistic logic, which is also called exact semantics, since it is based on a relation of exact verification between states and formulas. A natural question arises as to what are the limits of Fine’s approach and whether an exact semantics of similar kind can be constructed for other important non-classical logics. In our paper, we will generalize Fine’s approach and develop an exact semantics for some substructural logics. In particular, (...)
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  15. 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 for (...)
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  16. Weak relevant justification logics.Shawn Standefer - 2023 - Journal of Logic and Computation 33 (7):1665–1683.
    This paper will develop ideas from [44]. We will generalize their work in two directions. First, we provide axioms for justification logics over the base logic B and show that the logic permits a proof of the internalization theorem. Second, we provide alternative frames that more closely resemble the standard versions of the ternary relational frames, as well as a more general approach to the completeness proof. We prove that soundness and completeness hold for justification logics over a wide variety (...)
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  17. Against Classical Paraconsistent Metatheory.Koji Tanaka & Patrick Girard - 2023 - Analysis 83 (2):285-294.
    There was a time when 'logic' just meant classical logic. The climate is slowly changing and non-classical logic cannot be dismissed off-hand. However, a metatheory used to study the properties of non-classical logic is often classical. In this paper, we will argue that this practice of relying on classical metatheories is problematic. In particular, we will show that it is a bad practice because the metatheory that is used to study a non-classical logic often rules out the very logic it (...)
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  18. On the Origins of Gaggle Theory.Katalin Bimbo - 2022 - In Igor Sedlár (ed.), The Logica Yearbook, 2021. College Publications. pp. 19-38.
  19. A truth-maker semantics for ST: refusing to climb the strict/tolerant hierarchy.Ulf Hlobil - 2022 - Synthese 200 (5):1-23.
    The paper presents a truth-maker semantics for Strict/Tolerant Logic (ST), which is the currently most popular logic among advocates of the non-transitive approach to paradoxes. Besides being interesting in itself, the truth-maker presentation of ST offers a new perspective on the recently discovered hierarchy of meta-inferences that, according to some, generalizes the idea behind ST. While fascinating from a mathematical perspective, there is no agreement on the philosophical significance of this hierarchy. I aim to show that there is no clear (...)
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  20. Depth Relevance and Hyperformalism.Shay Allen Logan - 2022 - Journal of Philosophical Logic 51 (4):721-737.
    Formal symptoms of relevance usually concern the propositional variables shared between the antecedent and the consequent of provable conditionals. Among the most famous results about such symptoms are Belnap’s early results showing that for sublogics of the strong relevant logic R, provable conditionals share a signed variable between antecedent and consequent. For logics weaker than R stronger variable sharing results are available. In 1984, Ross Brady gave one well-known example of such a result. As a corollary to the main result (...)
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  21. Non-classical Comparative Logic I: Standard Categorical Logic–from SLe to IFLe.Amer Amikhteh & Seyed Ahmad Mirsanei - 2021 - Logical Studies 12 (1):1-24.
    n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules of (...)
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  22. Theorems of Alternatives for Substructural Logics.Almudena Colacito, Nikolaos Galatos & George Metcalfe - 2021 - In Ofer Arieli & Anna Zamansky (eds.), Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Springer Verlag. pp. 91-105.
    A theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative formulas in the “R-mingle” logic RM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {RM}$$\end{document} to the validity of a linear combination of these formulas, and Gordan’s theorem for solutions of linear systems over the real numbers that yields an analogous reduction for validity (...)
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  23. IKTω and Lukasiewicz-models.Andreas Fjellstad & Jan-Fredrik Olsen - 2021 - Notre Dame Journal of Formal Logic 62 (2):247 - 256.
    In this note, we show that the first-order logic IKω is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKTω obtained by expanding IKω with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this (...)
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  24. Hyperdoctrines and the Ontology of Stratified Semantics.Shay Logan - 2021 - In Davide Fazio, Antonio Ledda & Francesco Paoli (eds.), Algebraic Perspectives on Substructural Logics. Springer International Publishing. pp. 169-193.
    I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find (...)
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  25. Strong Depth Relevance.Shay Allen Logan - 2021 - Australasian Journal of Logic 18 (6):645-656.
    Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these (...)
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  26. On Not Saying What We Shouldn't Have to Say.Shay Logan & Leach-Krouse Graham - 2021 - Australasian Journal of Logic 18 (5):524-568.
    In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.
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  27. Pomset Logic: the other approach to non commutativity in logic.Christian Retoré - 2021 - In Claudia Casadio & Philip J. Scott (eds.), Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics. Springer Verlag. pp. 299-345.
    Thirty years ago, I introduced a noncommutative variant of classical linear logic, called pomset logic, coming from a particular categorical interpretation of linear logic known as coherence spaces. In addition to the usual commutative multiplicative connectives of linear logic, pomset logic includes a noncommutative connective, “⊲” called before, associative and self-dual: ⊥ = A⊥ ⊲ B⊥. The conclusion of a pomset logic proof is a Partially Ordered Multiset of formulas. Pomset logic enjoys a proof net calculus with cut-elimination, denotational semantics, (...)
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  28. Translating Metainferences Into Formulae: Satisfaction Operators and Sequent Calculi.Ariel Jonathan Roffé & Federico Pailos - 2021 - Australasian Journal of Logic 3.
    In this paper, we present a way to translate the metainferences of a mixed metainferential system into formulae of an extended-language system, called its associated σ-system. To do this, the σ-system will contain new operators (one for each standard), called the σ operators, which represent the notions of "belonging to a (given) standard". We first prove, in a model-theoretic way, that these translations preserve (in)validity. That is, that a metainference is valid in the base system if and only if its (...)
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  29. An Incompleteness Theorem for Modal Relevant Logics.Shawn Standefer - 2021 - Notre Dame Journal of Formal Logic 62 (4):669 - 681.
    In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.
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  30. The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework).Damian E. Szmuc - 2021 - Bulletin of the Section of Logic 50 (4):421-453.
    We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined (...)
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  31. Confused Entailment.Tore Fjetland Øgaard - 2021 - Topoi 41 (1):207-219.
    Priest argued in Fusion and Confusion (Priest in Topoi 34(1):55–61, 2015a) for a new concept of logical consequence over the relevant logic B, one where premises my be “confused” together. This paper develops Priest’s idea. Whereas Priest uses a substructural proof calculus, this paper provides a Hilbert proof calculus for it. Using this it is shown that Priest’s consequence relation is weaker than the standard Hilbert consequence relation for B, but strictly stronger than Anderson and Belnap’s original relevant notion of (...)
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  32. From Hilbert proofs to consecutions and back.Tore Fjetland Øgaard - 2021 - Australasian Journal of Logic 18 (2):51-72.
    Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus (...)
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  33. A Hierarchy of Classical and Paraconsistent Logics.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Journal of Philosophical Logic 49 (1):93-120.
    In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...)
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  34. A recovery operator for nontransitive approaches.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Review of Symbolic Logic 13 (1):80-104.
    In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...)
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  35. (I can’t get no) antisatisfaction.Pablo Cobreros, Elio La Rosa & Luca Tranchini - 2020 - Synthese 198 (9):8251-8265.
    Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...)
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  36. Metainferential duality.Bruno Da Ré, Federico Pailos, Damian Szmuc & Paula Teijeiro - 2020 - Journal of Applied Non-Classical Logics 30 (4):312-334.
    The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collect...
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  37. 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 been intensively studied over (...)
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  38. A Family of Strict/Tolerant Logics.Melvin Fitting - 2020 - Journal of Philosophical Logic 50 (2):363-394.
    Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences are treated more tolerantly. More specifically, ST is a three-valued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does. A version of this result has been extended to meta, metameta, … (...)
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  39. Expressing Validity: Towards a Self-Sufficient Inferentialism.Ulf Hlobil - 2020 - In Martin Blicha & Igor Sedlár (eds.), The Logica Yearbook 2019. London: College Publications. pp. 67-82.
    For semantic inferentialists, the basic semantic concept is validity. An inferentialist theory of meaning should offer an account of the meaning of "valid." If one tries to add a validity predicate to one's object language, however, one runs into problems like the v-Curry paradox. In previous work, I presented a validity predicate for a non-transitive logic that can adequately capture its own meta-inferences. Unfortunately, in that system, one cannot show of any inference that it is invalid. Here I extend the (...)
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  40. Putting the Stars in their Places.Shay Allen Logan - 2020 - Thought: A Journal of Philosophy 9 (3):188-197.
    This paper presents a new semantics for the weak relevant logic DW that makes the role of the infamous Routley star more explicable. Central to this rewriting is combining aspects of both the American and Australian plan for understanding negations in relevance logics.
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  41. Varieties of de Morgan monoids: Covers of atoms.T. Moraschini, J. G. Raftery & J. J. Wannenburg - 2020 - Review of Symbolic Logic 13 (2):338-374.
    The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may (...)
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  42. Non-contractability and Revenge.Julien Murzi & Lorenzo Rossi - 2020 - Erkenntnis 85 (4):905-917.
    It is often argued that fully structural theories of truth and related notions are incapable of expressing a nonstratified notion of defectiveness. We argue that recently much-discussed non-contractive theories suffer from the same expressive limitation, provided they identify the defective sentences with the sentences that yield triviality if they are assumed to satisfy structural contraction.
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  43. Tolerance and the distributed sorites.Zach Barnett - 2019 - Synthese 196 (3):1071-1077.
    On some accounts of vagueness, predicates like “is a heap” are tolerant. That is, their correct application tolerates sufficiently small changes in the objects to which they are applied. Of course, such views face the sorites paradox, and various solutions have been proposed. One proposed solution involves banning repeated appeals to tolerance, while affirming tolerance in any individual case. In effect, this solution rejects the reasoning of the sorites argument. This paper discusses a thorny problem afflicting this approach to vagueness. (...)
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  44. (Meta)inferential levels of entailment beyond the Tarskian paradigm.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2019 - Synthese 198 (S22):5265-5289.
    In this paper we discuss the extent to which the very existence of substructural logics puts the Tarskian conception of logical systems in jeopardy. In order to do this, we highlight the importance of the presence of different levels of entailment in a given logic, looking not only at inferences between collections of formulae but also at inferences between collections of inferences—and more. We discuss appropriate refinements or modifications of the usual Tarskian identity criterion for logical systems, and propose an (...)
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  45. Graham Priest on Dialetheism and Paraconsistency.Can Başkent & Thomas Macaulay Ferguson (eds.) - 2019 - Cham, Switzerland: Springer Verlag.
    This book presents the state of the art in the fields of formal logic pioneered by Graham Priest. It includes advanced technical work on the model and proof theories of paraconsistent logic, in contributions from top scholars in the field. Graham Priest’s research has had a considerable influence on the field of philosophical logic, especially with respect to the themes of dialetheism—the thesis that there exist true but inconsistent sentences—and paraconsistency—an account of deduction in which contradictory premises do not entail (...)
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  46. Stoic Sequent Logic and Proof Theory.Susanne Bobzien - 2019 - History and Philosophy of Logic 40 (3):234-265.
    This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic (...)
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  47. Faithfulness for naive validity.Ulf Hlobil - 2019 - Synthese 196 (11):4759-4774.
    Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. (...)
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  48. Notes on Stratified Semantics.Shay Allen Logan - 2019 - Journal of Philosophical Logic 48 (4):749-786.
    In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature, 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71, 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are, 27–59, 1988; Mares, Studia Logica 51, 1–20, 1992), both of which (...)
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  49. On elimination of quantifiers in some non‐classical mathematical theories.Guillermo Badia & Andrew Tedder - 2018 - Mathematical Logic Quarterly 64 (3):140-154.
    Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.
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  50. 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 clear (...)
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