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 an empirical-experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operationalrealistic tradition of Jauch and Piron, i. e. as an investigation of "the logic of yes-no experiments" (or "questions"). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the "non-classicality" of quantum behavior than any perspective based on static Propositional Logic. (shrink)

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.

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 the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2. (shrink)

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 →.

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. The argument is conducted via a thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices (3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic of bivalence realism (i.e., of truth and falsity), although it is consistent with some form of ontological realism (4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks. (shrink)

This chapter is offered as a contribution to the logic of down below. We attempt to demonstrate that the nature of human agency necessitates that there actually be such a logic. The ensuing sections develop the suggestion that cognition down below has a structure strikingly similar to the physical structure of quantum states. In its general form, this is not an idea that originates with the present authors. It is known that there exist mathematical models from the (...) cognitive science of cognition down below that have certain formal similarities to quantum mechanics. We want to take this idea seriously. We will propose that the subspaces of von Neumann-Birkhoff lattices are too crisp for modelling requisite cognitive aspects in relation to subsymbolic logic. Instead, we adopt an approach which relies on projections into nonorthogonal density states. The projection operator is motivated from cues which probe human memory. (shrink)

One of the most interesting programs in the foundations of quantum mechanics is the realist quantum logic approach associated with Putnam, Bub, Demopoulos and Friedman (and which is the focus of my own research.) I believe that realist quantum logic is our best hope for making sense of quantum mechanics, but I have come to suspect that the usual version may not be the correct one. In this paper, I would like to say why (...) and to propose an alternative. (shrink)

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 concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a “Bohrified” proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann. (shrink)

This paper gives a formulation of quantum logic in the abstract algebraic setting laid out by Dunn and Hardegree (2001). On this basis, it provides a comparative analysis of viable quantum logical bivalent semantics and their classical counterparts, thereby showing that the truth-functional status of classical and quantum connectives is not as different as usually thought. Then it points out that bivalent semantics for quantum logic - compatible with realism about quantum mechanics - (...) can be maintained, albeit at the price of truth-functionality. Finally, the paper improves on Hellman (1980)'s argument that this lack of truth-functionality entails a change of meaning between classical and quantum logical connectives. (shrink)

Is the quantum-logic interpretation dead? Its near total absence from current discussions about the interpretation of quantum theory suggests so. While mathematical work on quantum logic continues largely unabated, interest in the quantum-logic interpretation seems to be almost nil, at least in Anglo-American philosophy of physics. This paper has the immodest purpose of changing that fact. I shall argue that while the quantum-logic interpretation faces challenges, it remains a live option. The (...) usual objections either miss the mark, or admit a reasonable answer, or fail to decide the issue conclusively. (shrink)

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.

In 1936, G. Birkhoff and J. v. Neumann published an article with the title The logic of quantum mechanics'. In this paper, the authors demonstrated that in quantum mechanics the most simple observables which correspond to yes-no propositions about a quantum physical system constitute an algebraic structure, the most important proper ties of which are given by an orthocomplemented and quasimodular lattice Lq. Furthermore, this lattice of quantum mechanical proposi tions has, from a formal point (...) of view, many similarities with a Boolean lattice L8 which is known to be the lattice of classical propositional logic. Therefore, one could conjecture that due to the algebraic structure of quantum mechanical observables a logical calculus Q of quantum mechanical propositions is established, which is slightly different from the calculus L of classical propositional logic but which is applicable to all quantum mechanical propositions. This calculus has sometimes been called 'quan tum logic'. However, the statement that propositions about quantum physical systems are governed by the laws of quantum logic, which differ from ordinary classical logic and which are based on the empirically well-established quantum theory, is exposed to two serious objec tions: Logic is a theory which deals with those relationships between various propositions that are valid independent of the content of the respective propositions. Thus, the validity of logical relationships is not restricted to a special type of proposition, e. g. to propositions about classical physical systems. (shrink)

Is the quantum-logic interpretation dead? Its near total absence from current discussions about the interpretation of quantum theory suggests so. While mathematical work on quantum logic continues largely unabated, interest in the quantum-logic interpretation seems to be almost nil, at least in Anglo-American philosophy of physics. This paper has the immodest purpose of changing that fact. I shall argue that while the quantum-logic interpretation faces challenges, it remains a live option. The (...) usual objections either miss the mark, or admit a reasonable answer, or fail to decide the issue conclusively. (shrink)

These are the proceedings of the Workshop on Quantum Logic held in Erice (Sicily), December 2 - 9, 1979, at the Ettore Hajorana Centre for Scientific Culture. A conference of this sort was originally proposed by Giuliano Toraldo di Francia, who suggested the idea to Antonino Zichichi, and thus laid the foundation for the Workshop. To both of them we express our appreciation and thanks, also on behalf of the other participants, for having made this conference possible. There (...) were approximately fifty participants; their names and institutions are listed in the text. Quantum logic, which has now a history of some forty or more years, has seen remarkable growth during the sixties and seventies. The papers in the present volume presuppose, by and large, some acquaintance with the elements of the subject. These may be found in the well-known books by J.H. Jauch (Foundations of Quantum Hechanics; Reading, 1968), V.S. Varadarajan (Geometry of Quantum Theory: Princeton, 1968), and C. Piron (Foundations of Quantum Theory; New York, 1976). The initial program for the conference listed about twenty-five invited papers. But in the context of a very active and qualified attendance, other contributions were offered. This volume contains all of them. The program listed six main topics: I. Classification or different areas of quantum logic, and open problems. II. Comparison and unification of different approaches to quantum theories; problems of interpretation. III. Formal quantum logic; axiomatics. IV. Hodal interpretations of quantum logic. v vi FOREWORD V. Quantum set theory. (shrink)

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual (...) properties of physical systems. This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann (...) "quantum logic" can be interpreted by taking their "disjunction" to be ¬(∼A & ∼ B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added. (shrink)

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, 1976). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics. (shrink)

Starting with a quantum logic (a σ-orthomodular poset) L. a set of probabilistically motivated axioms is suggested to identify L with a standard quantum logic L(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space.

This book compares various approaches to the interpretation of quantum mechanics, in particular those which are related to the key words "the Copenhagen interpretation", "the antirealist view", "quantum logic" and "hidden variable theory". Using the concept of "correlation" carefully analyzed in the context of classical probability and in quantum theory, the author provides a framework to compare these approaches. He also develops an extension of probability theory to construct a local hidden variable theory. The book should (...) be of interest for physicists and philosophers of science interested in the foundations of quantum theory. (shrink)

In a recent paper, Michael Friedman and Hilary Putnam argued that the Luders rule is ad hoc from the point of view of the Copenhagen interpretation but that it receives a natural explanation within realist quantum logic as a probability conditionalization rule. Geoffrey Hellman maintains that quantum logic cannot give a non-circular explanation of the rule, while Jeffrey Bub argues that the rule is not ad hoc within the Copenhagen interpretation. As I see it, all four (...) are wrong. Given that there is to be a projection postulate, there are at least two natural arguments which the Copenhagen advocate can offer on behalf of the Luders rule, contrary to Friedman and Putnam. However, the argument which Bub offers is not a good one. At the same time, contrary to Hellman, quantum logic really does provide an explanation of the Luders rule, and one which is superior to that of the Copenhagen account, since it provides an understanding of why there should be a projection postulate at all. (shrink)

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 role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ , of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects. (shrink)

The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.

Is the quantum-logic interpretation dead? Its near total absence from current discussions about the interpretation of quantum theory suggests so. While mathematical work on quantum logic continues largely unabated, interest in the quantum-logic interpretation seems to be almost nil, at least in Anglo-American philosophy of physics. This paper has the immodest purpose of changing that fact. I shall argue that while the quantum-logic interpretation faces challenges, it remains a live option. The (...) usual objections either miss the mark, or admit a reasonable answer, or fail to decide the issue conclusively. (shrink)

Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truth-functionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering ("implication") to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially (...) ordered set of propositions as a logic--i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way. (shrink)

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 subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets. (shrink)

In quantum logics, the notions of strong and full order determination and unitality for states on orthomodular posets are well known. These notions are defined for hypergraphs and their state spaces in a consistent manner and the relations between them and to the notions defined for orthomodular posets are discussed. The state space of a hypergraph is a polytope. This polytope is a simplex if and only if every superposition of pure states is a mixture of these same pure (...) states. Isomorphic hypergraphs have convexly isomorphic state spaces. A class of hypergraphs is given whose group of automorphisms is group-isomorphic to the group of convex automorphisms of their state spaces. (shrink)

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus Teff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the 'value-definiteness' of propositions is not postulated, the calculus $T_{eff}$ represents a calculus of effective (intuitionistic) quantum logic. Beginning with the tableaux-calculus the equivalence of $T_{eff}$ to calculi which use more familiar figures such as sequents and implications can be investigated. (...) In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to $T_{eff}$ . The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space. (shrink)

Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.

In this work we discuss logical structures related to indistinguishable particles. Most of the framework used to develop these structures was presented in [17, 28] and in [20, 14, 15, 16]. We use these structures and constructions to discuss possible ontologies for identical particles. In other words, we use these structures in order to characterize the logical structure of quantum systems for the case of indistinguishable particles, and draw possible philosophical implications. We also review some proposals available in the (...) literature which may be considered within the framework of the quantum logical tradition regarding the problem of indistinguishability. Besides these discussions and constructions, we advance novel technical results, namely, a lattice theoretical structure for identical particles for the finite dimensional case. This kind of approach was not present in the scarcely literature of quantum logic and indistinguishable particles. (shrink)

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 for quantum logic which reveals its difference from classical logic in a strikingly intuitive way, viz. through the fact that quantum logic admits (suitably formulated versions of) the characteristic quantum-mechanical notions of superposition and incompatibility of attributes. That is, precisely the features that distinguish quantum from classical physics also serve, within this framework, to distinguish quantum from classical logic. Some light is shed on the question of whether quantum logic is a genuine logical system by introducing a natural entailment relation for quantum-logical formulas with the implication symbol. The novelty is that, although implication behaves as it should (i.e. the 'deduction theorem' holds), the order of introduction of premises is significant. The fact that a reasonable entailment relation can be formulated for quantum logic supports the view that it is a genuine logical system and not merely an algebraic formalism. (shrink)

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 of this analysis to concepts could alleviate some well known problems in cognitive science. (shrink)

We implement a recent characterization of metaphysical indeterminacy in the context of orthodox quantum theory, developing the syntax and semantics of two propositional logics equipped with determinacy and indeterminacy operators. These logics, which extend a novel semantics for standard quantum logic that accounts for Hilbert spaces with superselection sectors, preserve different desirable features of quantum logic and logics of indeterminacy. In addition to comparing the relative advantages of the two, we also explain how each (...) class='Hi'>logic answers Williamson’s challenge to any substantive account of determinacy: For any proposition p, what could the difference between “p” and “it’s determinate that p” ever amount to? (shrink)

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 and natural extension of Boolean logic, a subset which corresponds to incomplete statements. The philosophical background of this unsatisfactory state of affairs is briefly explored. (shrink)

The aim of this paper is to show that the argument put forth by Bell and Hallett against Putnam's thesis regarding the invariance of meaning for quantum logical connectives is insufficient to establish their conclusion. By using an example from the causal theory of time, the paper shows how the condition they specify as relevant in cases of meaning variance in fact fails. As a result, the conclusion that negation undergoes a change of meaning in the quantum logical (...) case is left in doubt. The paper proceeds in three stages. First, a summary of Putnam's argument for the invariance of meaning in the case of quantum logical connectives is provided; this is followed by a review of the criticisms advanced against it by Bell and Hallett. Finally, it is shown how the main claim upon which their major criticism rests is unfounded. (shrink)

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application “towards dynamic quantum logic”, namely in describing the temporal indeterministic propagation of actual (...) properties of physical systems. This paper can as such by conceived as an addendum to “Quantum Logic in Intuitionistic Perspective” that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

Quantum logic as genuine non-classical logic provides no solution to the "paradoxes" of quantum mechanics. From the minimal condition that synonyms be substitutable salva veritate, it follows that synonymous sentential connectives be alike in point of truth-functionality. It is a fact of pure mathematics that any assignment Φ of (0, 1) to the subspaces of Hilbert space (dim. ≥ 3) which guarantees truth-preservation of the ordering and truth-functionality of QL negation, violates truth-functionality of QL ∨ and (...) $\wedge $ . Thus, from within both the classical framework and that of any QL that preserves elementary set theory, two distinct (nonsynonymous) sets of connectives are discernible. Classical derivations of QM paradoxes are all available unless the language of QM is not classically closed. Maintaining this requires a strong and selfdefeating verification theory of meaning, the philosophical cornerstone of the Copenhagen interpretation to which QL was to provide an alternative. (shrink)

In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.

In their seminal paper Birkhoff and von Neumann revealed the following dilemma:[ ] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.

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 (...) a set of sentences ϕ V that are verifiable (or testable) according to ${{\mathcal{T}}, }$ on which a weak complementation ⊥ is induced by ${{\mathcal{T}}. }$ The triple $(\phi_{V},\prec,^{\perp})$ is then the desired concrete logic. By applying this procedure we recover a classical logic and a standard quantum logic as concrete logics associated with classical and quantum mechanics, respectively. The latter result is obtained in a purely formal way, but it can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Hence quantum logic can be considered as a mathematical structure formalizing the properties of the notion of verification in quantum physics. This conclusion supports the general idea that some nonclassical logics can coexist without conflicting with classical logic (global pluralism) because they formalize metalinguistic notions that do not coincide with the notion of truth as correspondence but are not alternative to it either. (shrink)

One problem with assessing quantum logic is that there are considerable differences between its practitioners. In particular they offer different versions of the set of sentences which the logic governs. On some accounts the sentences involved describe events, on others they are ascriptions of properties. In this paper a framework is offered within which to discuss different quantum logical interpretations of quantum theory, and then the works of Jauch, Putnam, van Fraassen and Kochen are located (...) within it. (shrink)