Abstract
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), Piron (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.
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References
Aerts, D. (1981). The one and the many, towards a unification of the quantum and the classical description of one and many physical entities. PhD-thesis, Free University of Brussels.
Aerts, D. (1983). The description of one and many physical systems. In C. Gruber, C. Piron, T. M. TL’m, R. Weill (Eds.), Les fondements de la mŐcanique quantique, 25e cours de perfectionnement de l’Association Vaudoise des Chercheurs en Physique Montana, du 6 au 12 mars 1983 (pp. 63–148). Lausanne: l’A.V.C.P.
Amira H. B., Stubbe I. (1998) How quantales emerge by introducing induction within the operational approach. Helvetica Physica Acta 71: 554–572
Balbiani, P. (2007) Propositional Dynamic Logic. Online entry of the Stanford Encyclopedia of Philosophy
Baltag A., Smets S. (2005) Complete axiomatizations for quantum actions. International Journal of Theoretical Physics 44(12): 2267–2282
Baltag, A., & Smets, S. (2004). The Logic of Quantum Programs. In The proceedings of the 2nd international workshop on quantum programming languages (QPL 2004), TUCS general publication no. 33, Turku Center for Computer Science.
Baltag A., Smets S. (2006) LQP: The dynamic logic of quantum information. Mathematical Structures in Computer Science, Special Issue on Quantum Programming Languages 16(3): 491–525
Baltag, A., & Smets S. (2008). A dynamic—logical perspective on quantum behavior. In I. Douven, & L. Horsten (Eds.), Special issue: Applied logic in the philosophy of science. Studia logica, 89, 185–209.
Baltag, A., & Smets, S. (2010a). Quantum logic as a dynamic logic. In T. Kuipers, J. van Benthem, & H. Visser (Eds.), Synthese, 179(2), 285–306.
Baltag A., Smets S. (2010b) Correlated knowledge, an epistemic-logic view on quantum entanglement. International Journal of Theoretical Physics 49(12): 3005–3021
Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843. Reprinted in C. A. Hooker (Ed.). (1975). The logico-algebraic approach to quantum mechanics (Vol. 1, pp. 1–26). Dordrecht: D. Reidel Publishing Company.
Chang, H. (2009). Operationalism. In Stanford encyclopedia of philosophy.
Coecke B., Moore D.J., Smets S. (2004) Logic of dynamics & dynamics of logic; some paradigm examples. In: Rahman S., Symons J., Gabbay D.M., Van Bendegem J.P. (eds) Logic, epistemology and the unity of science. Kluwer, Dordrecht, pp 527–556
Coecke B., Moore D. J., Stubbe I. (2001) Quantaloids describing causation and propagation for physical properties. Foundations of Physics Letters 14: 357–367
Coecke B., Smets S. (2004) The Sasaki Hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes. International Journal of Theoretical Physics 43: 1705–1736
Coecke B., Stubbe I. (1999) On a duality of quantales emerging from an operational resolution. International Journal of Theoretical Physics 38: 3269–3281
Dalla Chiara M., Giuntini R., Greechie R. (2004) Reasoning in quantum theory. Kluwer Academic Pub, Dordrecht
Dishkant H. (1972) Semantics of the minimal logic of quantum mechanics. Studia Logica 30: 23–30
Foulis D. (1999) A half century of quantum logic ? What have we learned?. In: Aerts D., Pykacz J. (eds) Quantum structures and the nature of reality. Kluwer Acad. Pub., Dordrecht, pp 1–36
Foulis D. J., Randall C. H. (1972) Operational statistics I., basic concepts. Journal of Mathematical Physics 13: 1667–1675
Foulis D. J., Randall C. H. (1984) A note on misunderstandings of piron’s axioms for quantum mechanics. Foundations of Physics 14: 65–81
Garson, J. (2009). “Modal Logic”, online entry of the Stanford Encyclopedia of Philosophy.
Goldblatt R. (1974) Semantic analysis of orthologic. Journal of Philosophical Logic 3: 19–35
Goldblatt R. (1984) Orthomodularity is not elementary. The Journal of Symbolic Logic 49: 401–404
Gottwald, S. (2009). Many-valued logic. In Online entry of the Stanford encyclopedia of philosophy.
Harel D., Kozen D., Tiuryn J. (2000) Dynamic logic. MIT Press, Cambridge
Hodges, W. (2009). Logic and games. In Online entry of the Stanford encyclopedia of philosophy
Husimi K. (1937) Studies on the foundations of quantum mechanics I. Proceedings of Physico-Mathematical Society Japan 9: 766–778
Jauch J. M., Piron C. (1969) On the structure of quantal proposition systems. Helvetica Physica Acta 42: 842–848
Kalmbach G. (1983) Orthomodular lattices. Academic Press, NY
Loomis, L. (1955). The lattice theoretic background of the dimension theory of operator algebras. Memoirs of the American Mathematical Society No. 18.
Maeda S. (1955) Dimension functions on certain general lattices. Journal of Science of the Hiroshima University A 19: 211–237
Mayet R. (1998) Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms. International Journal of Theoretical Physics 37(1): 109–114
Moore D. J. (1999) On state spaces and property lattices. Studies in History and Philosophy of Modern Physics 30: 61–83
Piron, C. (1964). Axiomatique quantique (PhD-Thesis). Helvetica Physica Acta, 37, 439–468. Quantum axiomatics (M. Cole, Trans.). RB4 Technical memo 107/106/104, GPO Engineering Department, London.
Piron, C. (1972). Survey of general quantum physics. Foundations of physics, 2, 287–314. Reprinted in C. A. Hooker (Ed.). (1975). The logico-algebraic approach to quantum mechanics (Vol. I). Dordrecht: D. Reidel Publishing Company.
Piron C. (1976) Foundations of quantum physics. W.A. Benjamin Inc, Massachusetts
Putnam H. (1968) Is logic empirical?. In: Cohen R., Wartofsky M. (eds) Boston studies in the philosophy of science. D. Reidel, Holland, Dordrecht
Randall C. H., Foulis D. J. (1973) Operational statistics II, manuals of operations and their logics. Journal of Mathematical Physics 14: 1472–1480
Smets S. (2006) From intuitionistic logic to dynamic operational quantum logic. Poznan Studies in Philosophy and the Humanities 91: 257–275
Smets, S. (2011). Logic and quantum physics, Journal of the Indian Council of Philosophical Research.
Solèr M. P. (1995) Characterization of Hilbert spaces by orthomodular spaces. Communications in Algebra 23(1): 219–243
van Benthem J. (1996) Exploring logical dynamics, studies in logic, language and information. CSLI Publications, Stanford
van Benthem, J. (2003). A mini-guide to logic in action.Research Reports PP-2004-02, University of Amsterdam
van Benthem J. (2010) Dynamics of information and interaction. Cambridge University Press, New York
von Neumann, J. (1932). Grundlagen der Quantenmechanik. Berlin: Springer Verlag (English translation: Mathematical foundations of quantum mechanics. New Jersey: Princeton University Press, 1996).
Wilce, A. (2002). Quantum logic and probability theory. In stanford encyclopedia of philosophy.
Wilce A. (2009) Test spaces. In: Engesser K., Gabbay D., Lehmann D. (eds) Handbook of quantum logic and quantum structures. North-Holland, Elsevier, pp 443–550
Acknowledgments
We thank Johan van Benthem for his valuable comments on this paper. Sonja Smets thanks the Netherlands Organization for Scientific Research for sponsoring this work under the VIDI research programme with number 639.072.904.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baltag, A., Smets, S. The dynamic turn in quantum logic. Synthese 186, 753–773 (2012). https://doi.org/10.1007/s11229-011-9915-7
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DOI: https://doi.org/10.1007/s11229-011-9915-7