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  1. Chess Composition as an Art.Miro Brada - manuscript
    The article presents the chess composition as a logical art, with concrete examples. It began with Arabic mansuba, and later evolved to new-strategy designed by Italian Alberto Mari. The redefinition of mate (e.g. mate with a free field) or a theme to quasi-pseudo theme, opens the new space for combinations, and enables to connect it with other fields like computer science. The article was exhibited in Holland Park, W8 6LU, The Ice House between 18. Oct - 3. Nov. 2013.
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  2. On Classical and Quantum Logical Entropy.David Ellerman - manuscript
    The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...)
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  3. On the Duality Between Existence and Information.David Ellerman - manuscript
    Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...)
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  4. Partitions and Objective Indefiniteness.David Ellerman - manuscript
    Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...)
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  5. Heisenberg Quantum Mechanics, Numeral Set-Theory And.Han Geurdes - manuscript
    In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical probability (...)
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  6. The Quantum Logic of Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.David Ellerman - 2017
    Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...)
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  7. Generalized Fuzzy Quantum Logics.J. Pykacz - forthcoming - Foundations of Physics.
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  8. Modal-Type Orthomodular Logic.Trixie Wagner & Andreas Schouml - forthcoming - Mathematical Logic Quarterly.
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  9. Universal Logic in Terms of Quantum Information.Vasil Penchev - 2020 - Metaphilosophy eJournal (Elsevier: SSRN) 12 (9):1-5.
    Any logic is represented as a certain collection of well-orderings admitting or not some algebraic structure such as a generalized lattice. Then universal logic should refer to the class of all subclasses of all well-orderings. One can construct a mapping between Hilbert space and the class of all logics. Thus there exists a correspondence between universal logic and the world if the latter is considered a collection of wave functions, as which the points in Hilbert space can be interpreted. The (...)
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  10. Pan demos Alla ricerca di un nuovo pensiero.Donato Santarcangelo - 2020 - Dialoghi Mediterranei.
  11. Categoricity and Possibility. A Note on Williamson's Modal Monism.Iulian D. Toader - 2020 - In The Logica Yearbook 2019. London: College Publications. pp. 221-231.
    The paper sketches an argument against modal monism, more specifically against the reduction of physical possibility to metaphysical possibility. The argument is based on the non-categoricity of quantum logic.
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  12. The Quantum Logic of Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.David Ellerman - 2018 - Logic Journal of the IGPL 26 (1):1-13.
    ince the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition is dual to the notion of a subset. Hence the Boolean logic of (...)
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  13. Topos Theoretic Quantum Realism.Benjamin Eva - 2017 - British Journal for the Philosophy of Science 68 (4):1149-1181.
    ABSTRACT Topos quantum theory is standardly portrayed as a kind of ‘neo-realist’ reformulation of quantum mechanics.1 1 In this article, I study the extent to which TQT can really be characterized as a realist formulation of the theory, and examine the question of whether the kind of realism that is provided by TQT satisfies the philosophical motivations that are usually associated with the search for a realist reformulation of quantum theory. Specifically, I show that the notion of the quantum state (...)
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  14. The Logic of the Future in Quantum Theory.Anthony Sudbery - 2017 - Synthese 194 (11):4429-4453.
    According to quantum mechanics, statements about the future made by sentient beings like us are, in general, neither true nor false; they must satisfy a many-valued logic. I propose that the truth value of such a statement should be identified with the probability that the event it describes will occur. After reviewing the history of related ideas in logic, I argue that it gives an understanding of probability which is particularly satisfactory for use in quantum mechanics. I construct a lattice (...)
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  15. Quantum Logic.C. de Ronde, G. Domenech & H. Freytes - 2016 - Internet Encyclopedia of Philosophy.
    Quantum Logic in Historical and Philosophical Perspective Quantum Logic was developed as an attempt to construct a propositional structure that would allow for describing the events of interest in Quantum Mechanics. QL replaced the Boolean structure, which, although suitable for the discourse of classical physics, was inadequate for representing the atomic realm. The … Continue reading Quantum Logic →.
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  16. Modality and Contextuality in Topos Quantum Theory.Benjamin Eva - 2016 - Studia Logica 104 (6):1099-1118.
    Topos quantum theory represents a whole new approach to the formalization of non-relativistic quantum theory. It is well known that TQT replaces the orthomodular quantum logic of the traditional Hilbert space formalism with a new intuitionistic logic that arises naturally from the topos theoretic structure of the theory. However, it is less well known that TQT also has a dual logical structure that is paraconsistent. In this paper, we investigate the relationship between these two logical structures and study the implications (...)
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  17. Quantum Logic Associated to Finite Dimensional Intervals of Modular Ortholattices.R. Giuntini, H. Freytes & G. Sergioli - 2016 - Journal of Symbolic Logic 81 (2):629-640.
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  18. A New View of Effects in a Hilbert Space.Roberto Giuntini, Antonio Ledda & Francesco Paoli - 2016 - Studia Logica 104 (6):1145-1177.
    We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.
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  19. First-Order Frames for Orthomodular Quantum Logic.Chrysafis Hartonas - 2016 - Journal of Applied Non-Classical Logics 26 (1):69-80.
    One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames we propose form a natural subclass of the orthoframes, where is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset. Our frame class contains the canonical orthomodular frame of the (...)
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  20. On the Notion of Truth in Quantum Mechanics: A Category-Theoretic Standpoint.Vassilios Karakostas & Elias Zafiris - 2016 - In Diederik Aerts, Christian de Ronde, Hector Freytes & Roberto Giuntini (eds.), Probing the Meaning and Structure of Quantum Mechanics: Semantics, Dynamics and Identity. World Scientific. pp. 1-43.
    The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valua- tion in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event al- gebras. We show explicitly that the latter category is equipped with an object of (...)
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  21. The Intrinsic Quantum Nature of Nash Equilibrium Mixtures.Yohan Pelosse - 2016 - Journal of Philosophical Logic 45 (1):25-64.
    In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...)
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  22. Loops, Projective Invariants, and the Realization of the Borromean Topological Link in Quantum Mechanics.Elias Zafiris - 2016 - Quantum Studies: Mathematics and Foundations 3 (4):337-359.
    All the typical global quantum mechanical observables are complex relative phases obtained by interference phenomena. They are described by means of some global geometric phase factor, which is thought of as the “memory” of a quantum system undergoing a “cyclic evolution” after coming back to its original physical state. The origin of a geometric phase factor can be traced to the local phase invariance of the transition probability assignment in quantum mechanics. Beyond this invariance, transition probabilities also remain invariant under (...)
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  23. Boolean Localization Of Quantum Events: A Processual Sheaf-Theoretic Approach.Elias Zafiris - 2016 - In David Ray Griffin, Michael Epperson & Timothy E. Eastman (eds.), Physics and Speculative Philosophy: Potentiality in Modern Science. De Gruyter. pp. 107-126.
  24. Logics of Informational Interactions.Alexandru Baltag & Sonja Smets - 2015 - Journal of Philosophical Logic 44 (6):595-607.
    The pre-eminence of logical dynamics, over a static and purely propositional view of Logic, lies at the core of a new understanding of both formal epistemology and the logical foundations of quantum mechanics. Both areas appear at first sight to be based on purely static propositional formalisms, but in our view their fundamental operators are essentially dynamic in nature. Quantum logic can be best understood as the logic of physically-constrained informational interactions between subsystems of a global physical system. Similarly, epistemic (...)
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  25. 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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  26. Quantum Team Logic and Bell’s Inequalities.Tapani Hyttinen, Gianluca Paolini & Jouko Väänänen - 2015 - Review of Symbolic Logic 8 (4):722-742.
    A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this (...)
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  27. The Lattice Structure of the S-Lorenz Core.Vincent Iehlé - 2015 - Theory and Decision 78 (1):141-151.
    For any TU game and any ranking of players, the set of all preimputations compatible with the ranking, equipped with the Lorenz order, is a bounded join semi-lattice. Furthermore, the set admits as sublattice the S-Lorenz core intersected with the region compatible with the ranking. This result uncovers a new property about the structure of the S-Lorenz core. As immediate corollaries, we obtain complementary results to the findings of Dutta and Ray :403–422, 1991), by showing that any S-constrained egalitarian allocation (...)
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  28. Primacy of Quantum Logic in the Natural World.Cynthia Sue Larson - 2015 - Cosmos and History 11 (2):326-340.
    72 1024x768 This paper presents evidence from the fields of cognitive science and quantum information theory suggesting quantum theory to be the dominant fundamental logic in the natural world, in direct challenge to the long-held assumption that quantum logic only need be considered ‘in the quantum realm.' A summary of the evolution of quantum logic and quantum theory is presented, along with an overview for the necessity of incomplete quantum knowledge, and some representative aspects of quantum logic. A case can (...)
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  29. Dynamical Correspondence in a Generalized Quantum Theory.Gerd Niestegge - 2015 - Foundations of Physics 45 (5):525-534.
    In order to figure out why quantum physics needs the complex Hilbert space, many attempts have been made to distinguish the C*-algebras and von Neumann algebras in more general classes of abstractly defined Jordan algebras . One particularly important distinguishing property was identified by Alfsen and Shultz and is the existence of a dynamical correspondence. It reproduces the dual role of the selfadjoint operators as observables and generators of dynamical groups in quantum mechanics. In the paper, this concept is extended (...)
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  30. Quantum Logic and Quantum Reconstruction.Allen Stairs - 2015 - Foundations of Physics 45 (10):1351-1361.
    Quantum logic understood as a reconstruction program had real successes and genuine limitations. This paper offers a synopsis of both and suggests a way of seeing quantum logic in a larger, still thriving context.
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  31. In the Chaos of Today's Society: The Dynamics of Collapse as Another Shift in the Quantum Anthropology of Heidi Ann Russell.Radek Trnka - 2015 - Prague: Togga.
    The presented study introduces a new theoretical model of collapse for social, cultural, or political systems. Based on the current form of quantum anthropology conceptualized by Heidi Ann Russell, further development of this field is provided. The new theoretical model is called the spiral model of collapses, and is suggested to provide an analytical framework for collapses in social, cultural, and political systems. The main conclusions of this study are: 1) The individual crises in the period before a collapse of (...)
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  32. Non-Reflexive Logical Foundation for Quantum Mechanics.Newton C. A. da Costa & Christian de Ronde - 2014 - Foundations of Physics 44 (12):1369-1380.
    On the one hand, non-reflexive logics are logics in which the principle of identity does not hold in general. On the other hand, quantum mechanics has difficulties regarding the interpretation of ‘particles’ and their identity, also known in the literature as ‘the problem of indistinguishable particles’. In this article, we will argue that non-reflexive logics can be a useful tool to account for such quantum indistinguishability. In particular, we will provide a particular non-reflexive logic that can help us to analyze (...)
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  33. Quantum Decoherence: A Logical Perspective.Sebastian Fortin & Leonardo Vanni - 2014 - Foundations of Physics 44 (12):1258-1268.
    The so-called classical limit of quantum mechanics is generally studied in terms of the decoherence of the state operator that characterizes a system. This is not the only possible approach to decoherence. In previous works we have presented the possibility of studying the classical limit in terms of the decoherence of relevant observables of the system. On the basis of this approach, in this paper we introduce the classical limit from a logical perspective, by studying the way in which the (...)
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  34. The New Quantum Logic.Robert B. Griffiths - 2014 - Foundations of Physics 44 (6):610-640.
    It is shown how all the major conceptual difficulties of standard (textbook) quantum mechanics, including the two measurement problems and the (supposed) nonlocality that conflicts with special relativity, are resolved in the consistent or decoherent histories interpretation of quantum mechanics by using a modified form of quantum logic to discuss quantum properties (subspaces of the quantum Hilbert space), and treating quantum time development as a stochastic process. The histories approach in turn gives rise to some conceptual difficulties, in particular the (...)
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  35. The Problem of Identity and a Justification for a Non-Reflexive Quantum Mechanics.D. Krause - 2014 - Logic Journal of the IGPL 22 (2):186-205.
    In this paper we try to justify our way of looking for an alternative approach to quantum mechanics, which is based on a non-classical logic. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. We characterize individuals, and then explain in what sense entanglement is a concept which can be applied to individuals in a restricted sense only. Then, we turn to indiscernibility and, after realizing that this concept is of a fundamental importance, (...)
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  36. A Generalized Quantum Theory.Gerd Niestegge - 2014 - Foundations of Physics 44 (11):1216-1229.
    In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus (which is a quantum logical model of the Lüders-von Neumann (...)
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  37. Model Theory of a Hilbert Space Expanded with an Unbounded Closed Selfadjoint Operator.Camilo Enrique Argoty Pulido - 2014 - Mathematical Logic Quarterly 60 (6):403-424.
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  38. The Quantum Harmonic Oscillator as a Zariski Geometry.Vinesh Solanki, Dmitry Sustretov & Boris Zilber - 2014 - Annals of Pure and Applied Logic 165 (6):1149-1168.
    A structure is associated with the quantum harmonic oscillator, over a fixed algebraically closed field FF of characteristic 0, which is shown to be uncountably categorical. An analysis of definable sets is carried out, from which it follows that this structure is a Zariski geometry of dimension 1. It is non-classical in the sense that it is not interpretable in ACF0ACF0 and in the case F=CF=C, is not a structure on a complex manifold.
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  39. Relational Hidden Variables and Non-Locality.Samson Abramsky - 2013 - Studia Logica 101 (2):411-452.
    We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.
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  40. La mecánica cuántica y la conceptualidad: materia, historias, semántica y espacio-tiempo.Diederik Aerts - 2013 - Scientiae Studia 11 (1):75-99.
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  41. Distribution in the Logic of Meaning Containment and in Quantum Mechanics.Ross T. Brady & Andrea Meinander - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 223--255.
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  42. The Paraconsistent Logic of Quantum Superpositions.Newton C. A. da Costa & Christian de Ronde - 2013 - Foundations of Physics 43 (7):845-858.
    Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, (...)
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  43. The Toffoli-Hadamard Gate System: An Algebraic Approach.Maria Luisa Dalla Chiara, Antonio Ledda, Giuseppe Sergioli & Roberto Giuntini - 2013 - Journal of Philosophical Logic 42 (3):467-481.
    Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. The basic algebraic properties of this system have been studied in Dalla Chiara et al. (Foundations of Physics 39(6):559–572, 2009), where we have introduced the notion of Shi-Aharonov quantum computational structure. In this paper we propose an algebraic abstraction from the Hilbert-space quantum computational structures, by introducing the notion of Toffoli-Hadamard algebra. From an intuitive point of (...)
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  44. Editors' Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing. [REVIEW]J. Michael Dunn, Lawrence S. Moss & Zhenghan Wang - 2013 - Journal of Philosophical Logic 42 (3):443-459.
  45. Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature.Michael Epperson & Elias Zafiris - 2013 - Lexington Books.
    Foundations of Relational Realism presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. -/- If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive (...)
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  46. Quantum Computational Logic with Mixed States.Hector Freytes & Graciela Domenech - 2013 - Mathematical Logic Quarterly 59 (1-2):27-50.
    In this paper we solve the problem how to axiomatize a system of quantum computational gates known as the Poincaré irreversible quantum computational system. A Hilbert-style calculus is introduced obtaining a strong completeness theorem.
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  47. Recovering Quantum Logic Within an Extended Classical Framework.Claudio Garola & Sandro Sozzo - 2013 - Erkenntnis 78 (2):399-419.
    We present a procedure which allows us to recover classical and nonclassical logical structures as concrete logics associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory ${{\mathcal{T}}}$ and classical language ${{\fancyscript{L}}}$ expressing ${{\mathcal{T}}, }$ an observative sublanguage L of ${{\fancyscript{L}}}$ with a notion of truth as correspondence, introducing in L a derived and theory-dependent notion of C-truth (true with certainty), defining a physical preorder $\prec$ induced by C-truth, and finally selecting (...)
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  48. Search for Quantum Reality.Stan Gudder - 2013 - Journal of Philosophical Logic 42 (3):525-533.
    We summarize a recent search for quantum reality. The full anhomomorphic logic of coevents for an event set is introduced. The quantum integral over an event with respect to a coevent is defined. Reality filters such as preclusivity and regularity of coevents are considered. A quantum measure that can be represented as a quantum integral with respect to a coevent is said to 1-generate that coevent. This gives a stronger filter that may produce a unique coevent called the “actual reality” (...)
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  49. Daggers, Kernels, Baer *-Semigroups, and Orthomodularity.John Harding - 2013 - Journal of Philosophical Logic 42 (3):535-549.
    We discuss issues related to constructing an orthomodular structure from an object in a category. In particular, we consider axiomatics related to Baer *-semigroups, partial semigroups, and various constructions involving dagger categories, kernels, and biproducts.
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  50. Weakly Intuitionistic Quantum Logic.Ronnie Hermens - 2013 - Studia Logica 101 (5):901-913.
    In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the proposal, namely, that propositions can always be seen as projections. The result is a weakly Heyting algebra for (...)
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