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  1. The Distributivity on Bi-Approximation Semantics.Tomoyuki Suzuki - 2016 - Notre Dame Journal of Formal Logic 57 (3):411-430.
    In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity—by the splitters and by the adjointness—can be explicated with the help of the axiom (...)
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  • Modal Definability in Enriched Languages.Valentin Goranko - 1989 - Notre Dame Journal of Formal Logic 31 (1):81-105.
    The paper deals with polymodal languages combined with standard semantics defined by means of some conditions on the frames. So, a notion of "polymodal base" arises which provides various enrichments of the classical modal language. One of these enrichments, viz. the base £(R,-R), with modalities over a relation and over its complement, is the paper's main paradigm. The modal definability (in the spirit of van Benthem's correspondence theory) of arbitrary and ~-elementary classes of frames in this base and in some (...)
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  • LK, LJ, Dual Intuitionistic Logic, and Quantum Logic.Hiroshi Aoyama - 2004 - Notre Dame Journal of Formal Logic 45 (4):193-213.
    In this paper, we study the relationship among classical logic, intuitionistic logic, and quantum logic . These logics are related in an interesting way and are not far apart from each other, as is widely believed. The results in this paper show how they are related with each other through a dual intuitionistic logic . Our study is completely syntactical.
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  • Paraconsistency Everywhere.Greg Restall - 2002 - Notre Dame Journal of Formal Logic 43 (3):147-156.
    “Paraconsistent” means “beyond the consistent” [3, 15]. Paraconsistent logics tolerate inconsistencies in a way that traditional logics do not. In a paraconsistent logic, the inference of explosion A, ∼AB is rejected. This may be for any of a number of reasons [16]. For proponents of relevance [1, 2] the argument has gone awry when we infer an irrelevant B from the inconsistent premises. Those who argue that inconsistent theories may have some logical content but do not commit us to everything, (...)
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  • The Impossibility of Vagueness.Kit Fine - 2008 - Philosophical Perspectives 22 (1):111-136.
    I wish to present a proof that vagueness is impossible. Of course, vagueness is possible; and so there must be something wrong with the proof. But it is far from clear where the error lies and, indeed, all of the assumptions upon which the proof depends are ones that have commonly been accepted. This suggests that we may have to radically alter our current conception of vagueness if we are to make proper sense of what it is.
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  • Computational Complexity for Bounded Distributive Lattices with Negation.Dmitry Shkatov & C. J. Van Alten - 2021 - Annals of Pure and Applied Logic 172 (7):102962.
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  • Valuations: Bi, Tri, and Tetra.Rohan French & David Ripley - 2019 - Studia Logica 107 (6):1313-1346.
    This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.
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  • Modal Translation of Substructural Logics.Chrysafis Hartonas - 2020 - Journal of Applied Non-Classical Logics 30 (1):16-49.
    ABSTRACTIn an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for BCI and stronger systems, but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural logic is a fragment of a corresponding sorted, residuated modal logic. At the conceptual and philosophical level, (...)
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  • HYPE: A System of Hyperintensional Logic.Hannes Leitgeb - 2019 - Journal of Philosophical Logic 48 (2):305-405.
    This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...)
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  • Lattice Logic as a Fragment of (2-Sorted) Residuated Modal Logic.Chrysafis Hartonas - 2019 - Journal of Applied Non-Classical Logics 29 (2):152-170.
    ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...)
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  • Valuations: Bi, Tri, and Tetra.Rohan French & David Ripley - 2019 - Studia Logica 107 (6):1313-1346.
    This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.
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  • On the Modal Logic of the Non-Orthogonality Relation Between Quantum States.Shengyang Zhong - 2018 - Journal of Logic, Language and Information 27 (2):157-173.
    It is well known that the non-orthogonality relation between the states of a quantum system is reflexive and symmetric, and the modal logic \ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies each (...)
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  • Correspondence Between Kripke Frames and Projective Geometries.Shengyang Zhong - 2018 - Studia Logica 106 (1):167-189.
    In this paper we show that some orthogeometries, i.e. projective geometries each defined using a ternary collinearity relation and equipped with a binary orthogonality relation, which are extensively studied in mathematics and quantum theory, correspond to Kripke frames, each defined using a binary relation, satisfying a few conditions. To be precise, we will define four special kinds of Kripke frames, namely, geometric frames, irreducible geometric frames, complete geometric frames and quantum Kripke frames; and we will show that they correspond to (...)
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  • Negation on the Australian Plan.Francesco Berto & Greg Restall - 2019 - Journal of Philosophical Logic 48 (6):1119-1144.
    We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. (...)
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  • Algorithmic Correspondence and Canonicity for Non-Distributive Logics.Willem Conradie & Alessandra Palmigiano - 2019 - Annals of Pure and Applied Logic 170 (9):923-974.
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  • Basis Logic for Application in Physics and Its Intuitionistic Alternative.Paul Weingartner - 2010 - Foundations of Physics 40 (9-10):1578-1596.
    This article proposes a basic logic for application in physics dispensing with the Principle of Excluded Middle. It is based on the article “Matrix Based Logics for Application in Physics (RMQ) which appeared 2009. In his article with Stachow on the Principle of Excluded Middle in Quantum Logic (QL), Peter Mittelstaedt showed that for some suitable QLs, including their own, the Principle of Excluded Middle can be added without any harm for QL; where ‘without any harm for QL’ means that (...)
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  • Matrix-Based Logic for Application in Physics.Paul Weingartner - 2009 - Review of Symbolic Logic 2 (1):132-163.
    The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense from those which are or in the sense of allowing only the most (...)
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  • Autoreferential Semantics for Many-Valued Modal Logics.Zoran Majkic - 2008 - Journal of Applied Non-Classical Logics 18 (1):79-125.
    In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices is based on relative pseudo-complements as in intuitionistic logic. (...)
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  • Hegel's Dialectics as a Semantic Theory: An Analytic Reading.Francesco Berto - 2007 - European Journal of Philosophy 15 (1):19–39.
  • Conditioning by Minimizing Accessibility.Konstantinos Georgatos - 2010 - In Giacomo Bonanno, Benedikt Löwe & Wiebe Hoek (eds.), Logic and the Foundations of Game and Decision Theory €“ Loft 8. Springer Berlin Heidelberg. pp. 20-33.
    This paper presents an axiomatization of a class of set-theoretic conditional operators using minimization of the geodesic distance defined as the shortest path generated by the accessibility relation on a frame. The objective of this modeling is to define conditioning based on a notion of similarity generated by degrees of indistinguishability.
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  • Absolute Contradiction, Dialetheism, and Revenge.Francesco Berto - 2014 - Review of Symbolic Logic 7 (2):193-207.
    Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...)
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  • Quantum Logic as a Dynamic Logic.Alexandru Baltag & Sonja Smets - 2011 - Synthese 179 (2):285 - 306.
    We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear "no". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with (...)
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  • Άδύνατον and Material Exclusion 1.Francesco Berto - 2008 - Australasian Journal of Philosophy 86 (2):165 – 190.
    Philosophical dialetheism, whose main exponent is Graham Priest, claims that some contradictions hold, are true, and it is rational to accept and assert them. Such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC). But all the classic formulations of the LNC are, in a sense, not questioned by a typical dialetheist, since she is (cheerfully) required to accept them by her own theory. The goal of this paper is to develop a formulation of the (...)
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  • Belief Update Using Graphs.Konstantinos Georgatos - 2008 - In David Wilson & Chad H. Lane (eds.), FLAIRS 21. AAAI Press. pp. 649-654.
    The purpose of this paper is to introduce a form of update based on the minimization of the geodesic distance on a graph. We provide a characterization of this class using set- theoretic operators and show that such operators bijectively correspond to geodesic metrics. As distance is generated by distinguishability, our framework is appropriate in contexts where distance is generated by threshold, and therefore, when measurement is erroneous.
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  • Model-Theoretic Investigations Into Consequence Operation (Cn) in Quantum Logics: An Algebraic Approach.Piotr Wilczek - unknown
    In this paper, we present the fundamentals of the so-called algebraic approach to propositional quantum logics. We define the set of formulae describing quantum reality as a free algebra freely generated by the set of quantum proportional variables. We define the general notion of logic as a structural consequence operation. Next, we introduce the concept of logical matrices understood as a model of quantum logics.We give the definitions of two quantum consequence operations defined in these models.
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  • Constructible Models of Orthomodular Quantum Logics.Piotr Wilczek - unknown
    We continue in this article the abstract algebraic treatment of quantum sentential logics Wil. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic - AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logics. We also proved several equivalences between state property systems (Jauch-Piron-Aerts line of investigations) and AAL treatment of quantum logics (corollary 18 and 19). We show that there (...)
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  • A Modality Called ‘Negation’.Francesco Berto - 2015 - Mind 124 (495):761-793.
    I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...)
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  • Conservativity of Heyting Implication Over Relevant Quantification.Robert Goldblatt - 2009 - Review of Symbolic Logic 2 (2):310-341.
    It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence (...)
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  • Mixed Algebras and Their Logics.Ivo Düntsch, Ewa Orłowska & Tinko Tinchev - 2017 - Journal of Applied Non-Classical Logics 27 (3-4):304-320.
    We investigate complex algebras of the form arising from a frame where, and exhibit their abstract algebraic and logical counterparts.
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  • Grishin Algebras and Cover Systems for Classical Bilinear Logic.Robert Goldblatt - 2011 - Studia Logica 99 (1-3):203-227.
    Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, (...)
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  • Duality for the Logic of Quantum Actions.Jort M. Bergfeld, Kohei Kishida, Joshua Sack & Shengyang Zhong - 2015 - Studia Logica 103 (4):781-805.
    In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” :2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a Hilbert space. The other approach considers a “Piron lattice”, which characterizes the algebra of closed linear subspaces of a Hilbert space. We define categories of these two sorts of structures and show (...)
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  • Logical Consequence and the Paradoxes.Edwin Mares & Francesco Paoli - 2014 - Journal of Philosophical Logic 43 (2-3):439-469.
    We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, (...)
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  • Deduction, Ordering, and Operations in Quantum Logic.Normal D. Megill & Mladen Pavičić - 2002 - Foundations of Physics 32 (3):357-378.
    We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that (...)
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  • An Alternative Propositional Calculus for Application to Empirical Sciences.Paul Weingartner - 2010 - Studia Logica 95 (1-2):233 - 257.
    The purpose of the paper is to show that by cleaning Classical Logic (CL) from redundancies (irrelevances) and uninformative complexities in the consequence class and from too strong assumptions (of CL) one can avoid most of the paradoxes coming up when CL is applied to empirical sciences including physics. This kind of cleaning of CL has been done successfully by distinguishing two types of theorems of CL by two criteria. One criterion (RC) forbids such theorems in which parts of the (...)
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  • An Exact Philosophy of Inexactness.Michael Katz - 1984 - Topoi 3 (1):43-53.
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  • Fuzzy Intuitionistic Quantum Logics.Gianpiero Cattaneo, Maria L. Dalla Chiara & Roberto Giuntini - 1993 - Studia Logica 52 (3):419 - 442.
    Fuzzy intuitionistic quantum logics (called also Brouwer-Zadeh logics) represent to non standard version of quantum logic where the connective not is split into two different negation: a fuzzy-like negation that gives rise to a paraconsistent behavior and an intuitionistic-like negation. A completeness theorem for a particular form of Brouwer-Zadeh logic (BZL 3) is proved. A phisical interpretation of these logics can be constructed in the framework of the unsharp approach to quantum theory.
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  • Bets and Boundaries: Assigning Probabilities to Imprecisely Specified Events.Peter Milne - 2008 - Studia Logica 90 (3):425-453.
    Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation (...)
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  • The Logics of Orthoalgebras.Maria Luisa Dalla Chiara & Roberto Giuntini - 1995 - Studia Logica 55 (1):3-22.
  • Some Foundational Problems in Mathematics Suggested by Physics.Maria Luisa Dalla Chiara - 1985 - Synthese 62 (2):303 - 315.
  • Languages of Similarity.Sŀawomir Bugajski - 1983 - Journal of Philosophical Logic 12 (1):1-18.
  • Quantum Logic and Physical Modalities.M. L. Dalla Chiara - 1977 - Journal of Philosophical Logic 6 (1):391-404.
  • Strong Versus Weak Quantum Consequence Operations.Jacek Malinowski - 1992 - Studia Logica 51 (1):113 - 123.
    This paper is a study of similarities and differences between strong and weak quantum consequence operations determined by a given class of ortholattices. We prove that the only strong orthologics which admits the deduction theorem (the only strong orthologics with algebraic semantics, the only equivalential strong orthologics, respectively) is the classical logic.
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  • Quantum Logics and Lindenbaum Property.Roberto Giuntini - 1987 - Studia Logica 46 (1):17 - 35.
    This paper will take into account the Lindenbaum property in Orthomodular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibility of extending any semantically non-contradictory set of formulas to a semantically non-contradictory complete set of formulas. The main purpose of this paper is to show that both OQL (...)
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  • An Axiom System for Orthomodular Quantum Logic.Gary M. Hardegree - 1981 - Studia Logica 40 (1):1 - 12.
    Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculusOMC is proposed for the orthomodular-valid formulas.OMC is based on two primitive connectives — the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra ofOMC is constructed, and it is shown to be (...)
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  • What is Quantum Logic?Sławomir Bugajski - 1982 - Studia Logica 41 (4):311 - 316.
    The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.
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  • Logical Self Reference, Set Theoretical Paradoxes and the Measurement Problem in Quantum Mechanics.Maria Luisa Dalla Chiara - 1977 - Journal of Philosophical Logic 6 (1):331-347.
  • On A Neglected Path to Intuitionism.Ian Rumfitt - 2012 - Topoi 31 (1):101-109.
    According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute (...)
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  • Some Properties of Orthologics.Yutaka Miyazaki - 2005 - Studia Logica 80 (1):75-93.
    In this paper, we present three main results on orthologics. Firstly, we give a sufficient condition for an orthologic to have variable separation property and show that the orthomodular logic has this property. Secondly, we show that the class of modular orthologics has an infinite descending chain. Finally we show that there exists a continuum of orthologics.
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  • The Dynamic Turn in Quantum Logic.Alexandru Baltag & Sonja Smets - 2012 - Synthese 186 (3):753 - 773.
    In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the "dynamic turn" in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42: 842-848, 1969), Pirón (Foundations of Quantum Physics, (...)
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  • Game-Theoretic Semantics for Non-Distributive Logics.Chrysafis Hartonas - 2019 - Logic Journal of the IGPL 27 (5):718-742.
    We introduce game-theoretic semantics for systems without the conveniences of either a De Morgan negation, or of distribution of conjunction over disjunction and conversely. Much of game playing rests on challenges issued by one player to the other to satisfy, or refute, a sentence, while forcing him/her to move to some other place in the game’s chessboard-like configuration. Correctness of the game-theoretic semantics is proven for both a training game, corresponding to Positive Lattice Logic and for more advanced games for (...)
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