## 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).

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## 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