Abstract
The word proposition is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages ℒ*(x) and ℒ(x) with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of p-testable propositions within the Boolean lattice of all propositions associated with sentences of ℒ(x). Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (truth and verifiability according to QM, respectively). Moreover one can construct a quantum language ℒ TQ (x) whose Lindenbaum–Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.
Similar content being viewed by others
References
Bell, J.S.: On the Einstein–Podolski–Rosen paradox. Physics 1, 195–200 (1964)
Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966)
Beltrametti, E., Cassinelli, G.: The Logic of Quantum Mechanics. Addison–Wesley, Reading (1981)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)
Busch, P., Lahti, P.J., Mittelstaedt, P.: The Quantum Theory of Measurement. Springer, Berlin (1991)
Busch, P., Shimony, A.: Insolubility of the quantum measurement problem for unsharp observables. Stud. Hist. Philos. Mod. Phys. B 27, 397–404 (1996)
Dalla Chiara, M., Giuntini, R., Greechie, R.: Reasoning in Quantum Theory. Kluwer, Dordrecht (2004)
Garola, C.: Classical foundations of quantum logic. Int. J. Theor. Phys. 30, 1–52 (1991)
Garola, C.: Against ‘paradoxes’: a new quantum philosophy for quantum mechanics. In: Aerts, D., Pykacz, J. (eds.) Quantum Physics and the Nature of Reality, pp. 103–140. Kluwer, Dordrecht (1999)
Garola, C.: Objectivity versus nonobjectivity in quantum mechanics. Found. Phys. 30, 1539–1565 (2000)
Garola, C.: A simple model for an objective interpretation of quantum mechanics. Found. Phys. 32, 1597–1615 (2002)
Garola, C.: MGP versus Kochen–Specker condition in hidden variables theories. Int. J. Theor. Phys. 44, 807–814 (2005)
Garola, C., Pykacz, J.: Locality and measurements within the SR model for an objective interpretation of quantum mechanics. Found. Phys. 34, 449–475 (2004)
Garola, C., Solombrino, L.: The theoretical apparatus of semantic realism: A new language for classical and quantum physics. Found. Phys. 26, 1121–1164 (1996)
Garola, C., Solombrino, L.: Semantic realism versus EPR-like paradoxes: the Furry, Bohm–Aharonov and Bell paradoxes. Found. Phys. 26, 1329–1356 (1996)
Garola, C., Sozzo, S.: A semantic approach to the completeness problem in quantum mechanics. Found. Phys. 34, 1249–1266 (2004)
Garola, C., Sozzo, S.: On the notion of proposition in classical and quantum mechanics. In: Garola, C., Rossi, A., Sozzo, S. (eds.) The Foundations of Quantum Mechanics. Historical Analysis and Open Questions—Cesena 2004. World Scientific, Singapore (2006)
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)
Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, New York (1974)
Jauch, J.M.: Foundations of Quantum Mechanics. Addison–Wesley, Reading (1968)
Ludwig, G.: Foundations of Quantum Mechanics I. Springer, New York (1983)
Mermin, N.D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993)
Piron, C.: Foundations of Quantum Physics. Benjamin, Reading (1976)
Randall, C.H., Foulis, D.J.: Properties and operational propositions in quantum mechanics. Found. Phys. 13, 843–857 (1983)
Rédei, M.: Quantum Logic in Algebraic Approach. Kluwer, Dordrecht (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garola, C. Physical Propositions and Quantum Languages. Int J Theor Phys 47, 90–103 (2008). https://doi.org/10.1007/s10773-007-9372-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-007-9372-8