Hermann Dishkant. The first order predicate calculus based on the logic of quantum mechanics. Reports on mathematical logic, no. 3 (1974), pp. 9–17. - G. N. Georgacarakos. Orthomodularity and relevance. Journal of philosophical logic, vol. 8 (1979), pp. 415–432. - G. N. Georgacarakos. Equationally definable implication algebras for orthomodular lattices. Studia logica, vol. 39 (1980), pp. 5–18. - R. J. Greechie and S. P. Gudder. Is a quantum logic a logic?Helvetica physica acta, vol. 44 (1971), pp. 238–240. - Gary M. Hardegree. The conditional in abstract and concrete quantum logic. The logico-algehraic approach to quantum mechanics, volume II, Contemporary consolidation, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1979, pp. 49–108. - Gary M. Hardegree. Material implication in orthomodular (and Boolean) lattices. Notre Dame journal of formal logic, vol. 22 (1981), pp. 163–182. - J. M. Jauch and C. Piron. What is “quantum-logic”?Quanta, Essays in theoretical physics dedicated to Gregor Wentzel, edited by P. G. O. Freund, C. J. Goebel, and Y. Nambu, The University of Chicago Press, Chicago and London1970, pp. 166–181. - Jerzy Kotas. An axiom system for the modular logic. English with Polish and Russian summaries. Studia logica, vol. 21 (1967), pp. 17–38. - P. Mittelstaedt. On the interpretation of the lattice of subspaces of the Hilbert space as a propositional calculus. Zeitschrift für Naturforschung, vol. 27a no. 8–9 (1972), pp. 1358–1362. - J. Jay Zeman. Generalized normal logic. Journal of philosophical logic, vol. 7(1978), pp. 225–243.

1983 ◽  
Vol 48 (1) ◽  
pp. 206-208 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Propositional Calculus ◽  
Axiom System ◽  
Theoretical Physics ◽  
Reidel Publishing Company ◽  
Philosophical Logic ◽  
Orthomodular Lattices ◽  
Implication Algebras ◽  
The University

Simon Kochen and E. P. Specker. Logical structures arising in quantum theory. A reprint of XL 507(21). The logieo-algebraic approach to quantum mechanics, Volume I, Historicale evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 263–276. - Simon Kochen and E. P. Specker. The calculus of partial propositional functions. A reprint of XL 508(20). The logieo-algebraic approach to quantum mechanics, Volume I, Historical evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 277–292. - P.D. Finch. On the structure of quantum logic. The logieo-algebraic approach to quantum mechanics, Volume I, Historical evolution, edited by C. A. Hooker, The University of Western Ontario series in philosophy of science, vol. 5, D. Reidel Publishing Company, Dordrecht and Boston1975, pp. 415–425. (Reprinted from The journal of symbolic logic, vol. 34 (1969), pp. 275–282.) - Stanley P. Gudder. Partial algebraic structures associated with orthomodular posets. Pacific journal of mathematics, vol. 41 (1972), pp. 717–730. - Janusz Czelakowski. Logics based on partial Boolean σ-algebras (1). Studia logica, vol. 33 (1974), pp. 371–396. - Gary M. Hardegree and Patricia J. Frazer. Charting the labyrinth of quantum logics: a progress report. Current issues in quantum logic, edited by Enrico G. Beltrametti and Bas C. van Fraassen, Ettore Majorana international science series, physical sciences, vol. 8, Plenum Press, New York and London1981, pp. 53–76. - Janusz Czelakowski. Partial referential matrices for quantum logics. Current issues in quantum logic, edited by Enrico G. Beltrametti and Bas C. van Fraassen, Ettore Majorana international science series, physical sciences, vol. 8, Plenum Press, New York and London1981, pp. 131–146.

1985 ◽  
Vol 50 (2) ◽  
pp. 558-566 ◽  
Author(s):  
R. I. G. Hughes
Keyword(s):  
New York ◽  
Quantum Mechanics ◽  
Philosophy Of Science ◽  
Quantum Logic ◽  
Algebraic Approach ◽  
Publishing Company ◽  
Reidel Publishing Company ◽  
Quantum Logics ◽  
Ettore Majorana ◽  
The University

scholarly journals Classical Logic in the Quantum Context

2020 ◽  
Vol 2 (4) ◽  
pp. 600-616
Author(s):  
Andrea Oldofredi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Classical Logic ◽  
Theory Of Measurement ◽  
Propositional Calculus ◽  
Bohmian Mechanics ◽  
Physical Content ◽  
Present Essay ◽  
Classical Propositional Calculus

It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.

scholarly journals The logic induced by effect algebras

2020 ◽  
Vol 24 (19) ◽  
pp. 14275-14286 ◽  
Author(s):  
Ivan Chajda ◽  
Radomír Halaš ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Chain Condition ◽  
Axiom System ◽  
Single Element ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Implication Algebras ◽  
Lattice Effect ◽  
Ascending Chain Condition

Abstract Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras $${\mathbf {E}}$$ E , we investigate a natural implication and prove that the implication reduct of $${\mathbf {E}}$$ E is term equivalent to $${\mathbf {E}}$$ E . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

William L. Harper. A sketch of some recent developments in the theory of conditionals. Ifs, Conditionals, belief, decision, chance, and time, edited by William L. Harper, Robert Stalnaker, and Glenn Pearce, The University of Western Ontario series in philosophy of science, vol. 15, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1981, pp. 3–38. - Robert C. Stalnaker. A theory of conditionals. A reprint of XLVII 470. Ifs, Conditionals, belief, decision, chance, and time, edited by William L. Harper, Robert Stalnaker, and Glenn Pearce, The University of Western Ontario series in philosophy of science, vol. 15, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1981, pp. 41–55. - David Lewis. Counterfactuals and comparative possibility. Ifs, Conditionals, belief, decision, chance, and time, edited by William L. Harper, Robert Stalnaker, and Glenn Pearce, The University of Western Ontario series in philosophy of science, vol. 15, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1981, pp. 57–85. (Reprinted from Journal of philosophical logic, vol. 2 (1973), pp. 418–446; also reprinted in Contemporary research in philosophical logic and linguistic semantics, Proceedings of a conference held at the University of Western Ontario, London, Canada, edited by D. Hockney, W. Harper, and B. Freed, The University of Western Ontario series in philosophy of science, vol. 4, D. Reidel Publishing Company, Dordrecht and Boston 1975, pp. 1–29.) - Robert C. Stalnaker. A defense of conditional excluded middle. Ifs, Conditionals, belief, decision, chance, and time, edited by William L. Harper, Robert Stalnaker, and Glenn Pearce, The University of Western Ontario series in philosophy of science, vol. 15, D. Reidel Publishing Company, Dordrecht, Boston, and London, 1981, pp. 87–104.

1984 ◽  
Vol 49 (4) ◽  
pp. 1411-1413
Author(s):  
Barry Loewer
Keyword(s):  
Philosophy Of Science ◽  
David Lewis ◽  
Publishing Company ◽  
Reidel Publishing Company ◽  
Philosophical Logic ◽  
Contemporary Research ◽  
Recent Developments ◽  
The University ◽  
Excluded Middle

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

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