Abstract
The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either ‘true’ or ‘false’, describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of ‘no go’ theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen–Specker contradiction. In this respect, the Bub–Clifton ‘uniqueness theorem’ is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences.
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Notes
The principle of value-definiteness has variously been called in the literature as, for instance, “the determined value assumption” in Auletta (2001, pp. 21, 105).
In justifying from a physical point of view the aforementioned term, it is worthy to note that the state D A , which results as a listing of well-defined properties or equivalently determinate truth-value assignments selected by a 2-valued homomorphism on \( L_{H} \left( {\left\{ {D_{{A_{i} }} } \right\}} \right) \), may naturally be viewed as constituting a state preparation of system S in the context of the preferred observable A to be measured. Thus, the state D A should not be regarded as the final post-measurement state, reached after an A-measurement has been carried out on the system concerned. On the contrary, the contextual state represents here an alternative description of the initial state D of S by taking specifically into account the selection of a particular observable, and hence of a suitable experimental context, on which the state of the system under measurement can be conditioned. In other words, it provides a redescription of the measured system which is necessitated by taking specifically into account the context of the selected observable. For, it is important to realize that this kind of redescription is intimately related to the fact that both states D and D A represent the same object system S, albeit in different ways. Whereas D refers to a general initial state of S independently of the specification of any particular observable, and hence, regardless of the determination of any measurement context, the state D A constitutes a conditionalization state preparation of S with respect to the observable to be measured, while dropping all ‘unrelated’ reference to observables that are incompatible with such a preparation procedure.
In fact, the determinate observable A picks up a Boolean sublattice in \( L_{H} \left( {\left\{ {D_{{A_{i} }} } \right\}} \right) \) which, in view of the Bub-Clifton theorem, is straightforwardly extended to \( L_{H} \left( {\left\{ {D_{{A_{i} }} } \right\}} \right) \) itself. The latter comprises as determinate all observables whose eigenspaces are spanned by the rays \( L_{H} \left( {\left\{ {D_{{A_{i} }} } \right\}} \right) \), given the system’s state D and A. These technicalities, however, bear no further significance for present purposes.
It should be pointed out that Bohr already on the basis of his complementarity principle introduced the concept of a ‘quantum phenomenon’ to refer “exclusively to observations obtained under specified circumstances, including an account of the whole experiment” (Bohr 1963, p. 73). This feature of context-dependence is also present in Bohm’s ontological interpretation of quantum theory by clearly putting forward that “quantum properties cannot be said to belong to the observed system alone and, more generally, that such properties have no meaning apart from the total context which is relevant in any particular situation. In this sense, this includes the overall experimental arrangement so that we can say that measurement is context dependent” (Bohm and Hiley 1993, p. 108).
References
Auletta G (2001) Foundations and interpretation of quantum mechanics. World Scientific, Singapore
Blackburn S (1996) The Oxford dictionary of philosophy. Oxford University Press, Oxford
Bohm D (1952) A suggested interpretation of quantum theory in terms of “hidden variables”, parts I and II. Phys Rev 85:166–179, 180–193
Bohm D, Hiley B (1993) The undivided universe: an ontological interpretation of quantum theory. Routledge, London
Bohr N (1963) Essays 1958–1962 on atomic physics and human knowledge. Wiley, New York
Bonjour L (1985) The structure of empirical knowledge. Harvard University Press, Cambridge
Bub J (2009) Bub–Clifton theorem. In: Greenberger D, Hentschel K, Weinert F (eds) Compendium of quantum physics. Springer, Berlin, pp 84–86
Bub J, Clifton R (1996) A uniqueness theorem for “no collapse” interpretations of quantum mechanics. Stud Hist Phil Mod Phys 27:181–219
Burgess A, Burgess JP (2011) Truth. Princeton University Press, Princeton
Cabello A (2006) How many questions do you need to prove that unasked questions have no answers? Int J Quantum Inform 4:55–61
Candlish S, Damnjanovic N (2007) A brief history of truth. In: Jacquette D (ed) Philosophy of logic. Elsevier, Amsterdam, pp 227–323
Dalla Chiara M, Roberto G, Greechie R (2004) Reasoning in quantum theory. Kluwer, Dordrecht
Davis M (1977) A relativity principle in quantum mechanics. Int J Theor Phys 16:867–874
Devitt M (2001) The metaphysics of truth. In: Lynch M (ed) The nature of truth: classic and contemporary perspectives. MIT Press, Cambridge, pp 579–611
Dieks D, Vermaas P (eds) (1998) The modal interpretation of quantum mechanics. Kluwer, Dordrecht
Dirac PAM (1958) Quantum mechanics, 4th edn. Clarendon Press, Oxford
Engel P (2002) Truth. Acumen, Chesham
Greenberger D (2009) GHZ (Greenberger–Horne–Zeilinger) theorem and GHZ states. In: Greenberger D, Hentschel K, Weinert F (eds) Compendium of quantum physics. Springer, Berlin, pp 258–263
Karakostas V (2012) Realism and objectivism in quantum mechanics. J Gen Phil Sci 43:45–65
Kirchmair G, Zähringer F, Gerritsma R, Kleinmann M, Gühne O, Cabello A, Blatt R, Roos C (2009) State-independent experimental test of quantum contextuality. Nature 460:494–497
Kochen S, Specker E (1967) The problem of hidden variables in quantum mechanics. J Math Mech 17:59–87
Lewis D (2001) Forget about the ‘correspondence theory of truth’. Analysis 61:275–280
Lynch M (ed) (2001) The nature of truth: classic and contemporary perspectives. MIT Press, Cambridge
McDermid D (1998) Pragmatism and truth: the comparison objection to correspondence. Rev Metaphys 51:775–811
Rédei M (1998) Quantum logic in algebraic approach. Kluwer, Dordrecht
Svozil K (2009) Contexts in quantum, classical and partition logic. In: Engesser K, Gabbay D, Lehmann D (eds) Handbook of quantum logic and quantum structures: quantum logic. Elsevier, Amsterdam, pp 551–586
Takeuti G (1978) Two applications of logic to mathematics, part I: Boolean valued analysis. Publ Math Soc Japan 13, Iwanami and Princeton University Press, Tokyo and Princeton
Von Neumann J (1995) Mathematical foundations of quantum mechanics. Princeton University Press, Princeton
Yu S, Oh CH (2012) State-independent proof of Kochen–Specker theorem with 13 rays. Phys Rev Lett 108:030402
Zafiris E, Karakostas V (2013) A categorial semantic representation of quantum event structures. Found Phys 43:1090–1123
Acknowledgments
For discussion and comments on previous versions, I thank participants at audiences in the Seventh European Conference of Analytic Philosophy (Milan) and Fourteenth International Congress of Logic, Methodology and Philosophy of Science (Nancy). I also acknowledge support from the research programme ‘Thalis’ co-financed by the European Union (ESF) and the Hellenic Research Council (Project 70-3-11604).
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Karakostas, V. Correspondence Truth and Quantum Mechanics. Axiomathes 24, 343–358 (2014). https://doi.org/10.1007/s10516-013-9226-3
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DOI: https://doi.org/10.1007/s10516-013-9226-3