Abstract
In the orthodox quantum logic approach to foundations of quantum mechanics originated by Birkhoff and von Neumann [1] the object called quantum logic is not a logic in the very sense of this word but an algebraic structure: orthomodular lattice or poset. These orthomodular structures, abstracted from structures of Hilbert spaces used to describe quantum phenomena are weakenings of Boolean algebras. They can be thought of as representing Lindenbaum algebras of systems of propositions about quantum objects, the propositions belonging to the quantum logic proper. Since Lindenbaum algebras of propositional systems which obey the rules of classical logic are Boolean algebras it is inferred that the quantum logic proper, whatever it means, is non-classical.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Birkhoff, G., von Neumann, J.: 1936, ‘The Logic of Quantum Mechanics’, Ann Math, 37, 823–843.
Burmeister, P., Mg.czynski, M.: 1994, ‘Orthomodular (Partial) Algebras and their Representations’, Demonstratio Mathematica, 27, 701–722.
Foulis, D.J., Bennett, M.K.: 1994, ‘Effect Algebras and Unsharp Quantum Logics’, Found Phys, 24, 1331–1352.
Foulis, D.J., Greechie, R.J., Rüttimann, G.T.: 1992, ‘Filters and Supports in Orthoalgebras’, Int J Theor Phys, 37, 789–807.
Frink, 0.: 1938, ‘New Algebras of Logic’, Am Math Monthly, 45, 210–219.
Giles, R.: 1976, ‘Lukasiewicz Logic and Fuzzy Set Theory’, Int J Man-Machine Studies,8 313–327.
Giuntini, R., Greuling, H.: 1989, ‘Toward a Formal Language for Unsharp Properties’, Found Phys,19 931–945.
Lukasiewicz, J.: 1970, Selected Works. North-Holland, Amsterdam.
Mgczynski, M.J.: 1973, ‘The Orthogonality Postulate in Axiomatic Quantum Mechanics’, Int J Theor Phys, 8, 353–360.
Maczynski, M.J.: 1974, ‘Functional Properties of Quantum Logics’, Int J Theor Phys, 11, 149–156.
Mesiar, R.: 1994, ‘h-Fuzzy Quantum Logics’, Int J Theor Phys, 33, 1417–1425.
Mielnik, B.: 1976, Quantum Logic: Is it Necessarily Orthocomplemented?, in M. Flato et al., eds., Quantum Mechanics, Determinism, Causality, and Particles. Reidel, Dordrecht, 117–135.
Mittelstaedt, P., Prieur, A., Schieder, R.: 1987, ‘Unsharp Particle-Wave Duality in a Photon Split-Beam Experiment’, Found Phys, 17, 891–903.
Pykacz, J 1983, ‘Affine Maczynski Logics on Compact Convex Serts of States’, Int J Theor Phys, 22, 97–106.
Pykacz, J.: 1987, Quantum Logics as Families of Fuzzy Subsets of the Set of Physical States, Preprints of the Second International Fuzzy Systems Association Congress, Tokyo, July 20–25, 1987, Vol. 2, 437–440.
Pykacz, J.: 1987, ‘Quantum Logics and Soft Fuzzy Probability Spaces’, Bulletin pour les Sous-Ensembles Flous et leurs Applications, 32, 150–157.
Pykacz, J.: 1992, ‘Fuzzy Set Ideas in Quantum Logics’, Int J Theor Phys, 31, 1767–1783.
Pykacz, J.: 1994, ‘Fuzzy Quantum Logics and Infinite-Valued Lukasiewicz Logic’, Int J Theor Phys, 33, 1403–1416.
Pykacz, J., manuscript in preparation.
Wooters, W.K., Zurek, W.H.: 1979, ‘Complementation in the Double-Slit Experiment: Quantum Nonseparability and a Quantitative Statement of Bohr’s Principle’, Phys Rev D, 19, 473–484.
Zadeh, L.A.: 1965, ‘Fuzzy Sets’, Information and Control, 8, 338–353.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Pykacz, J. (1999). Attempt at the Logical Explanation of the Wave-Particle Duality. In: Chiara, M.L.D., Giuntini, R., Laudisa, F. (eds) Language, Quantum, Music. Synthese Library, vol 281. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2043-4_25
Download citation
DOI: https://doi.org/10.1007/978-94-017-2043-4_25
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5229-2
Online ISBN: 978-94-017-2043-4
eBook Packages: Springer Book Archive