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  1. Andrew Aberdein & Stephen Read (2009). The Philosophy of Alternative Logics. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. 613-723.
    This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...)
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  2. T. Acton, S. Caffrey, S. Dunn, P. Vinson & K. Svozil (1998). Analogues of Quantum Complementarity in the Theory of Automata - a Prolegomenon to the Philosophy of Quantum Mechanics. Studies in History and Philosophy of Science Part B 29 (1):61-80.
    Complementarity is not only a feature of quantum mechanical systems but occurs also in the context of finite automata.
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  3. Diederik Aerts (2013). La mecánica cuántica y la conceptualidad: materia, historias, semántica y espacio-tiempo. Scientiae Studia 11 (1):75-99.
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  4. Diederik Aerts, Ellie D'Hondt & Liane Gabora (2000). Why the Disjunction in Quantum Logic is Not Classical. Foundations of Physics 30 (9):1473-1480.
    The quantum logical `or' is analyzed from a physical perspective. We show that it is the existence of EPR-like correlation states for the quantum mechanical entity under consideration that make it nonequivalent to the classical situation. Specifically, the presence of potentiality in these correlation states gives rise to the quantum deviation from the classical logical `or'. We show how this arises not only in the microworld, but also in macroscopic situations where EPR-like correlation states are present. We investigate how application (...)
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  5. Constantin Antonopoulos (2007). The Quantum Logic of Zeno: Misconceptions and Restorations. Acta Philosophica 16 (2):265-284.
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  6. Hiroshi Aoyama (2004). LK, LJ, Dual Intuitionistic Logic, and Quantum Logic. Notre Dame Journal of Formal Logic 45 (4):193-213.
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  7. Michael Ashcroft (2010). Does Science Influence the Logic We Ought to Use: A Reflection on the Quantum Logic Controversy. Studia Logica 95 (1/2):183 - 206.
    In this article I argue that there is a sense in which logic is empirical, and hence open to influence from science. One of the roles of logic is the modelling and extending of natural language reasoning. It does so by providing a formal system which succeeds in modelling the structure of a paradigmatic set of our natural language inferences and which then permits us to extend this structure to novel cases with relative ease. In choosing the best system of (...)
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  8. I. C. Baianu (2006). Robert Rosen's Work and Complex Systems Biology. Axiomathes 16 (1-2):25-34.
    Complex Systems Biology approaches are here considered from the viewpoint of Robert Rosen’s (M,R)-systems, Relational Biology and Quantum theory, as well as from the standpoint of computer modeling. Realizability and Entailment of (M,R)-systems are two key aspects that relate the abstract, mathematical world of organizational structure introduced by Rosen to the various physicochemical structures of complex biological systems. Their importance for understanding biological function and life itself, as well as for designing new strategies for treating diseases such as cancers, is (...)
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  9. I. C. Baianu, R. Brown, G. Georgescu & J. F. Glazebrook (2006). Complex Non-Linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks. [REVIEW] Axiomathes 16 (1-2):65-122.
    A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil (...)
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  10. A. Baltag & S. Smets (2008). A Dynamic-Logical Perspective on Quantum Behavior. Studia Logica 89 (2):187 - 211.
    In this paper we show how recent concepts from Dynamic Logic, and in particular from Dynamic Epistemic logic, can be used to model and interpret quantum behavior. Our main thesis is that all the non-classical properties of quantum systems are explainable in terms of the non-classical flow of quantum information. We give a logical analysis of quantum measurements (formalized using modal operators) as triggers for quantum information flow, and we compare them with other logical operators previously used to model various (...)
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  11. Alexandru Baltag & Sonja Smets (2012). The Dynamic Turn in Quantum Logic. 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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  12. Alexandru Baltag & Sonja Smets (2011). Quantum Logic as a Dynamic Logic. 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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  13. Alexandru Baltag & Sonja Smets, The Logic of Quantum Programs.
    We present a logical calculus for reasoning about information flow in quantum programs. In particular we introduce a dynamic logic that is capable of dealing with quantum measurements, unitary evolutions and entanglements in compound quantum systems. We give a syntax and a relational semantics in which we abstract away from phases and probabilities. We present a sound proof system for this logic, and we show how to characterize by logical means various forms of entanglement (e.g. the Bell states) and various (...)
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  14. H. Barnum (2003). Quantum Information Processing, Operational Quantum Logic, Convexity, and the Foundations of Physics. Studies in History and Philosophy of Science Part B 34 (3):343-379.
    Quantum information science is a source of task-related axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform on a system: ''operational states.'' I discuss general frameworks for ''operational theories'' (sets of possible operational states of a system), in which convexity plays key role. The main technical content of the paper is in a theorem that any such theory naturally (...)
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  15. J. L. Bell (1986). A New Approach to Quantum Logic. British Journal for the Philosophy of Science 37 (1):83-99.
    The idea of a 'logic of quantum mechanics' or quantum logic was originally suggested by Birkhoff and von Neumann in their pioneering paper [1936]. Since that time there has been much argument about whether, or in what sense, quantum 'logic' can be actually considered a true logic (see, e.g. Bell and Hallett [1982], Dummett [1976], Gardner [1971]) and, if so, how it is to be distinguished from classical logic. In this paper I put forward a simple and natural semantical framework (...)
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  16. J. L. Bell (1985). Orthospaces and Quantum Logic. Foundations of Physics 15 (12):1179-1202.
    In this paper we construct the ortholattices arising in quantum logic starting from the phenomenologically plausible idea of a collection of ensembles subject to passing or failing various “tests.” A collection of ensembles forms a certain kind of preordered set with extra structure called anorthospace; we show that complete ortholattices arise as canonical completions of orthospaces in much the same way as arbitrary complete lattices arise as canonical completions of partially ordered sets. We also show that the canonical completion of (...)
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  17. John Bell & Michael Hallett (1982). Logic, Quantum Logic and Empiricism. Philosophy of Science 49 (3):355-379.
    This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong (1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation crucially does not. (...)
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  18. E. G. Beltrametti & G. Cassinelli (1977). On State Transformations Induced by Yes-No Experiments, in the Context of Quantum Logic. Journal of Philosophical Logic 6 (1):369 - 379.
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  19. E. G. Beltrametti & G. Cassinelli (1972). Quantum Mechanics Andp-Adic Numbers. Foundations of Physics 2 (1):1-7.
    We study the possibility of representing the proposition lattice associated with a quantum system by a linear vector space with coefficients from ap-adic field. We find inconsistencies if the lattice is assumed, as usual, to be irreducible, complete, orthocomplemented, atomic, and weakly modular.
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  20. Tomasz Bigaj (2001). Three-Valued Logic, Indeterminacy and Quantum Mechanics. Journal of Philosophical Logic 30 (2):97-119.
    The paper consists of two parts. The first part begins with the problem of whether the original three-valued calculus, invented by J. Łukasiewicz, really conforms to his philosophical and semantic intuitions. I claim that one of the basic semantic assumptions underlying Łukasiewicz's three-valued logic should be that if under any possible circumstances a sentence of the form "X will be the case at time t" is true (resp. false) at time t, then this sentence must be already true (resp. false) (...)
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  21. William Boos (1996). Mathematical Quantum Theory I: Random Ultrafilters as Hidden Variables. Synthese 107 (1):83 - 143.
    The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...)
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  22. J. Brabec & P. Pták (1982). On Compatibility in Quantum Logics. Foundations of Physics 12 (2):207-212.
    We offer a variant of the intrinsic definition of compatibility in logics. We shown that any compatible subset can be embedded into a Boolean σ-algebra, we show how the algebra is constructed, and we demonstrate that our definition cannot be weakened unless we put additional assumptions on the logic.
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  23. Joseph E. Brenner (2010). The Philosophical Logic of Stéphane Lupasco (1900–1988). Logic and Logical Philosophy 19 (3):243-285.
    The advent of quantum mechanics in the early 20 th Century had profound consequences for science and mathematics, for philosophy (Schrödinger), and for logic (von Neumann). In 1968, Putnam wrote that quantum mechanics required a revolution in our understanding of logic per se. However, applications of quantum logics have been little explored outside the quantum domain. Dummett saw some implications of quantum logic for truth, but few philosophers applied similar intuitions to epistemology or ontology. Logic remained a truth-functional ’science’ of (...)
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  24. T. A. Brody (1984). On Quantum Logic. Foundations of Physics 14 (5):409-430.
    The status and justification of quantum logic are reviewed. On the basis of several independent arguments it is concluded that it cannot be a logic in the philosophical sense of a general theory concerning the structure of valid inferences. Taken as a calculus for combining quantum mechanical propositions, it leaves a number of significant aspects of quantum physics unaccounted for. It is shown, moreover, that quantum logic, far from being more general than Boolean logic, forms a subset of a slight (...)
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  25. Jan Broekaert, Diederik Aerts & Bart D.’Hooghe (2006). The Generalised Liar Paradox: A Quantum Model and Interpretation. [REVIEW] Foundations of Science 11 (4):399-418.
    The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere (Aerts, Broekaert, & Smets, [Foundations of Science 1999, 4, 115–132; International Journal of Theoretical Physics, 2000, 38, 3231–3239]; Aerts and colleagues[Dialogue in Psychology, 1999, 10; Proceedings of Fundamental Approachs to Consciousness, Tokyo ’99; Mind in Interaction]. Basically, the intrinsic contextuality of the truth-value (...)
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  26. Jan Broekaert & Bart D'Hooghe (2000). A Model with Quantum Logic, but Non-Quantum Probability: The Product Test Issue. [REVIEW] Foundations of Physics 30 (9):1481-1501.
    We introduce a model with a set of experiments of which the probabilities of the outcomes coincide with the quantum probabilities for the spin measurements of a quantum spin- $ \frac{1}{2} $ particle. Product tests are defined which allow simultaneous measurements of incompatible observables, which leads to a discussion of the validity of the meet of two propositions as the algebraic model for conjunction in quantum logic. Although the entity possesses the same structure for the logic of its experimental propositions (...)
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  27. Jeffrey Bub (1994). How to Interpret Quantum Mechanics. Erkenntnis 41 (2):253 - 273.
    I formulate the interpretation problem of quantum mechanics as the problem of identifying all possible maximal sublattices of quantum propositions that can be taken as simultaneously determinate, subject to certain constraints that allow the representation of quantum probabilities as measures over truth possibilities in the standard sense, and the representation of measurements in terms of the linear dynamics of the theory. The solution to this problem yields a modal interpretation that I show to be a generalized version of Bohm's hidden (...)
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  28. Jeffrey Bub (1994). On the Structure of Quantal Proposition Systems. Foundations of Physics 24 (9):1261-1279.
    I define sublaltices of quantum propositions that can be taken as having determinate (but perhaps unknown) truth values for a given quantum state, in the sense that sufficiently many two-valued maps satisfying a Boolean homomorphism condition exist on each determinate sublattice to generate a Kolmogorov probability space for the probabilities defined by the slate. I show that these sublattices are maximal, subject to certain constraints, from which it follows easily that they are unique. I discuss the relevance of this result (...)
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  29. Jeffrey Bub (1982). Quantum Logic, Conditional Probability, and Interference. Philosophy of Science 49 (3):402-421.
    Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ , or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a (...)
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  30. Jeffrey Bub (1981). Hidden Variables and Quantum Logic — a Sceptical Review. Erkenntnis 16 (2):275 - 293.
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  31. Jeffrey Bub (1980). Book Review:Quantum Logic Peter Mittelstaedt. [REVIEW] Philosophy of Science 47 (2):332-.
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  32. Jeffrey Bub (1979). Some Reflections on Quantum Logic and Schrödinger's Cat. British Journal for the Philosophy of Science 30 (1):27-39.
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  33. Sŀawomir Bugajski (1978). Probability Implication in the Logics of Classical and Quantum Mechanics. Journal of Philosophical Logic 7 (1):95 - 106.
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  34. Sławomir Bugajski (1982). What is Quantum Logic? 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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  35. David Buhagiar, Emmanuel Chetcuti & Anatolij Dvurečenskij (2009). On Gleason's Theorem Without Gleason. Foundations of Physics 39 (6):550-558.
    The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In (...)
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  36. Philip G. Calabrese (2005). Toward a More Natural Expression of Quantum Logic with Boolean Fractions. Journal of Philosophical Logic 34 (4):363 - 401.
    This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due (...)
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  37. Cristian S. Calude, Peter H. Hertling & Karl Svozil (1999). Embedding Quantum Universes in Classical Ones. Foundations of Physics 29 (3):349-379.
    Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical ones, (...)
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  38. Martijn Caspers, Chris Heunen, Nicolaas P. Landsman & Bas Spitters (2009). Intuitionistic Quantum Logic of an N-Level System. Foundations of Physics 39 (7):731-759.
    A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...)
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  39. G. Cattaneo, R. Giuntini & S. Pulmannovà (2000). Pre-BZ and Degenerate BZ Posets: Applications to Fuzzy Sets and Unsharp Quantum Theories. [REVIEW] Foundations of Physics 30 (10):1765-1799.
    Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the so-called (...)
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  40. Gianpiero Cattaneo, Maria L. Dalla Chiara & Roberto Giuntini (1993). Fuzzy Intuitionistic Quantum Logics. 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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  41. Gianpiero Cattaneo & Roberto Giuntini (1995). Some Results on BZ Structures From Hilbertian Unsharp Quantum Physics. Foundations of Physics 25 (8):1147-1183.
    Some algebraic structures determined by the class σ(þ) of all effects of a Hilbert space þ and by some subclasses of σ(þ) are investigated, in particular de Morgan-Brouwer-Zadeh posets [it is proved that σ(þ n )(n<∞) has such a structure], Brouwer-Zadeh * posets (a quite trivial example consisting of suitable effects is given), and Brouwer-Zadeh 3 posets which are both de Morgan and *.It is shown that a nontrivial class of effects of a Hilbert space exists which is a BZ (...)
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  42. Gianpiero Cattaneo & Stanley Gudder (1999). Algebraic Structures Arising in Axiomatic Unsharp Quantum Physics. Foundations of Physics 29 (10):1607-1637.
    This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with an (...)
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  43. Gianpiero Cattaneo & Federico Laudisa (1994). Axiomatic Unsharp Quantum Theory (From Mackey to Ludwig and Piron). Foundations of Physics 24 (5):631-683.
    On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a “state-event-probability” (SEVP) structure, using a quite standard “fuzzification” procedure, a set of unsharp events (or “effects”) is constructed and the corresponding “state-effect-probability” (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an (...)
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  44. Gianpiero Cattaneo, Tiziana Marsico, Giuseppe Nisticò & Guido Bacciagaluppi (1997). A Concrete Procedure for Obtaining Sharp Reconstructions of Unsharp Observables in Finite-Dimensional Quantum Mechanics. Foundations of Physics 27 (10):1323-1343.
    We discuss the problem of how a (commutative) generalized observable in a finite-dimensional Hilbert space (communtative effect-valued resolution of the identity) can be considered as an unsharp realization of some standard observable (projection-valued resolution of the identity). In particular, we give a concrete procedure for constructing such a standard observable. Some results about the “uniqueness” of the resulting observable are also examined.
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  45. Ariadna Chernavska (1981). The Impossibility of a Bivalent Truth-Functional Semantics for the Non-Boolean Propositional Structures of Quantum Mechanics. Philosophia 10 (1-2):1-18.
    The general fact of the impossibility of a bivalent, truth-functional semantics for the propositional structures determined by quantum mechanics should be more subtly demarcated according to whether the structures are taken to be orthomodular latticesP L or partial-Boolean algebrasP A; according to whether the semantic mappings are required to be truth-functional or truth-functional ; and according to whether two-or-higher dimensional Hilbert spaceP structures or three-or-higher dimensional Hilbert spaceP structures are being considered. If the quantumP structures are taken to be orthomodular (...)
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  46. M. L. Dalla Chiara (1977). Quantum Logic and Physical Modalities. Journal of Philosophical Logic 6 (1):391 - 404.
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  47. M. L. Dalla Chiara & R. Giuntini (1994). Partial and Unsharp Quantum Logics. Foundations of Physics 24 (8):1161-1177.
    The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, (...)
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  48. Maria Luisa Dalla Chiara (1990). Review: Peter Gibbins, Particles and Paradoxes. The Limits of Quantum Logic. [REVIEW] Journal of Symbolic Logic 55 (1):354-355.
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  49. Maria Luisa Dalla Chiara & Roberto Giuntini (2000). Paraconsistent Ideas in Quantum Logic. Synthese 125 (1-2):55-68.
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  50. Maria Luisa Dalla Chiara & Roberto Giuntini (1989). Paraconsistent Quantum Logics. Foundations of Physics 19 (7):891-904.
    Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.
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