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MATRIX-BASED LOGIC FOR APPLICATION IN PHYSICS

Published online by Cambridge University Press:  01 March 2009

PAUL WEINGARTNER*
Affiliation:
Department Of Philosophy, University Of Salzburg
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF SALZBURG AND INSTITUT FÜR WISSENSCHAFTSTHEORIE SALZBURG, AUSTRIA, A-5020, E-mail:paul.weingartner@sbg.ac.at

Abstract

The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense “irrelevant” from those which are “relevant” or “nonredundant” in the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid in RMQ, whereas the former are only materially valid. RMQ is a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper how RMQ by its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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References

BIBLIOGRAPHY

Aerts, D. (1985). A possible explanation for the probabilities of quantum mechanics and example of a macroscopical system that violates Bell inequalities. In Mittelstaedt, P., and Stachow, E. W., editors. Recent Developments in Quantum Logics. Mannheim, Germany: B.I. Wissenschaftsverlag. pp. 235249.Google Scholar
Aerts, D., & Aerts, S. (2005). Towards a general operational and realistic framework for quantum mechanics and relativity theory. In Elitzur, A., Dolev, S., and Kolenda, N., editors. Quo Vadis Quantum Mechanics? Heidelberg, Germany: Springer, pp. 153207.CrossRefGoogle Scholar
Bell, J. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge, UK: Cambridge University Press.Google Scholar
Beltrametti, E. G., & Maczynski, M. J. (1991). On the characterization of classical and non-classical probabilities. Journal of Mathematical Physics, 32, 12801286.CrossRefGoogle Scholar
Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823843.CrossRefGoogle Scholar
Cattaneo, G., Dalla Chiara, M. L., & Giuntini, R. (2004). An unsharp quantum logic from quantum computation. In Weingartner, P., editor. Alternative Logics. Do Sciences Need Them? Heidelberg, Germany: Springer. pp. 323338.CrossRefGoogle Scholar
Czermak, J. (1981). Eine endliche Axiomatisierung von SS1M. In Morscher, E., Neumaier, O., and Zecha, G., editors. Philosophie als Wissenschaft/ [Essays in Scientific Philosophy]. Comes, Bad Reichenhall. pp. 245258.Google Scholar
Da Costa, N. (1974). On a theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497510.CrossRefGoogle Scholar
Dalla Chiara, M. L. (1977). Quantum logic and physical modalities. Journal of Philosohical Logic, 6, 391404.CrossRefGoogle Scholar
Dalla Chiara, M. L., & Guintini, R. (2001). Quantum Logics. arXiv: quant-ph/0101028 v2.Google Scholar
Dalla Chiara, M. L., Guintini, R., & Greechie, R. (2004). Reasoning in Quantum Theory, Sharp and Unsharp Quantum Logics. Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
D’Espagnat, B. (1979). A la Recherche du Réel. Paris, France: Gauthier-Villars.Google Scholar
Finch, P. D. (1970). Quantum logic as an implication algebra. Bulletin of the Australian Mathematical Society, 2, 101106.CrossRefGoogle Scholar
Finkelstein, D. (1979). Matter, space and logic. In Hooker, C. A., editor. The Logico-Algebraic Approach to Quantum Mechanics. Berlin: Springer, Vol. II. pp. 123139.CrossRefGoogle Scholar
Goldblatt, R. I. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 1935.CrossRefGoogle Scholar
Mittelstaedt, P. (1972). On the interpretation of the lattice of subspaces of Hilbert space as a propositional calculus. Zeitschrift für Naturforschung, 27a, 13581362.Google Scholar
Mittelstaedt, P. (1978). Quantum Logic. Dordrecht, The Netherlands: Reidel.CrossRefGoogle Scholar
Mittelstaedt, P. (1998). The Interpretation of Quantum Mechanics and the Measurement Process. Cambridge, UK: Cambridge University Press.Google Scholar
Mittelstaedt, P. (2004). Does quantum physics require a new logic? In Weingartner, P., editor. Alternative Logics. Do Sciences Need Them? Heidelberg, Germany: Springer. pp. 269284.CrossRefGoogle Scholar
Mittelstaedt, P., & Stachow, E. W., editors (1985). Recent Developments in Quantum Logics. Mannheim, Germany: B.I. Wissenschaftsverlag.Google Scholar
Mittelstaedt, P., & Weingartner, P. (2005). Laws of Nature. Heidelberg, Germany: Springer.Google Scholar
Piron, C. (1964). Axiomatique quantique. Helvetica Physica Acta, 37, 439468.Google Scholar
Pitowsky, I. (1989). Quantum Probability-Quantum Logic. Lecture Notes in Physics, Vol. 321. Berlin, Germany: Springer.Google Scholar
Priest, G. (2000). Motivation of paraconsistency: The slippery slope from classical logic to dialethism. In Batens, D., Morfensen, Ch., Priest, G., and van Bendegem, J. P., editors. Frontiers in Paraconsistent Logics. London: Research Studies Press, pp.223233.Google Scholar
Schurz, G. (1991a). Relevant deduction. Erkenntnis, 35, 391437.Google Scholar
Schurz, G. (1991b). Relevant deductive inference: Criteria and logics. In Schurz, G., and Dorn, G., editors. Advances in Scientific Philosophy. Essays in Honour of Paul Weingartner. Amsterdam, The Netherlands: Rodopi, pp. 5784.Google Scholar
Schurz, G., & Weingartner, P. (1987) Versimilitude defined by relevant consequence-elements. A new reconstruction of Popper’s original idea. In Kuipers, T., editor. What is Closer-to-the-Truth? Amsterdam, The Netherland: Rodopi, pp. 4777.Google Scholar
Schurz, G., & Weingartner, P. (2008). Zwart and Franssen’s impossibility theorem holds for possible-world-accounts but not for consequence-accounts to versimilitude. Synthese. Forthcoming.Google Scholar
Solèr, M. P. (1995). Characterisation of Hilbert spaces by orthomudular lattices. Communications in Algebra, 23(1), 219243.CrossRefGoogle Scholar
Stachow, E. W. (1976). Completeness of quantum logic. Journal of Philosophical Logic, 5, 237280.CrossRefGoogle Scholar
Stachow, E. W. (1978). Quantum logical calculi and lattice structures. Journal of Philosophical Logic, 7, 347386.CrossRefGoogle Scholar
Stachow, E. W. (2004). Experimental approach to quantum-logical connectives. In Weingartner, P., editor. Alternative Logics. Do Sciences Need Them? Heidelberg, Germany: Springer. pp. 285298.CrossRefGoogle Scholar
Takeuti, G. (1987). Proof Theory. Amsterdam: North Holland.Google Scholar
Tarski, A. (1956). Logic Semantics and Metamathematics. Oxford, UK: Oxford University Press.Google Scholar
Weingartner, P. (1968). Modal logics with two kinds of necessity and possibility. Notre Dame Journal of Formal Logic, 9(2), 97159.CrossRefGoogle Scholar
Weingartner, P. (1985). A simple relevance-criterion for natural language and its semantics. In Dorn, G., and Weingartner, P., editors. Foundations of Logic and Linguistics: Problems and their Solutions. New York: Plenum Press. pp. 563575.CrossRefGoogle Scholar
Weingartner, P. (2000a). Basic Questions on Truth (Series Episteme 24). Dordrecht, The Netherlands: Kluwer.CrossRefGoogle Scholar
Weingartner, P. (2000b). Reasons for filtering classical logic. In Batens, D., Morfensen, Ch., Priest, G., and van Bendegem, J. P., editors. Frontiers in Paraconsistent Logics. London: Research Studies Press, pp. 315327.Google Scholar
Weingarnter, P. (2001). Applications of logic outside logic and mathematics: Do such applications force us to deviate from classical logic?! In Stelzner, W., editor. Zwischen traditioneller und moderner Logik. Paderborn: Mentis, pp. 5364.Google Scholar
Weingartner, P., editor (2004a). Alternative Logics. Do Sciences Need Them? Heidelberg, Germany: Springer.CrossRefGoogle Scholar
Weingartner, P. (2004b). Reasons from science for limiting classical logic. In Weingartner, P., editor. Alternative Logics. Do Sciences Need Them? Heidelberg, Germany: Springer. pp. 233248.CrossRefGoogle Scholar
Weingartner, P., & Schurz, G. (1986). Paradoxes solved by simple relevance criteria. Logique et Analyse, 113, 340.Google Scholar
Zeilinger, A. (1992). Physik und Wirklichkeit. Neuere Entwicklungen zum Einstein-Podolsk-Rosen Paradoxon. In Reichel, H. C., and Prat de la Riba, E., editors. Naturwissenschaft und Weltbild. Vienna, Austria: Hölder-Pichler-Temsky, pp. 99121.Google Scholar
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