Abstract
Many scholars maintain that the language of quantum mechanics introduces a quantum notion of truth which is formalized by (standard, sharp) quantum logic and is incompatible with the classical (Tarskian) notion of truth. We show that quantum logic can be identified (up to an equivalence relation) with a fragment of a pragmatic language \(\mathcal {L}_{G}^{P}\) of assertive formulas, that are justified or unjustified rather than trueor false. Quantum logic can then be interpreted as an algebraic structure that formalizes properties of the notion of empirical justification according to quantum mechanics rather than properties of a quantum notion of truth. This conclusion agrees with a general integrationist perspective that interprets nonstandard logics as theories of metalinguistic notions different from truth, thus avoiding incompatibility with classical notions and preserving the globality of logic.
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Notes
We recall that modality operators apply to sentences of propositional or predicate logics, producing sentences of higher logical level that are either true orfalse (hence they extend CL). For instance, the alethic modality operator \(\square \) (necessarily) is interpreted as “it is necessary that...” and applies to sentences stating, e.g., relations among physical events. The assertion sign ⊩, instead, does not describe the act of assertion (i.e., it cannot be interpreted as “it is asserted that...”) but, as stated above, it shows such an act in the object language, producing assertions that are not true or false but justified or unjustified.
As an undecidable statement that is relevant for physics we recall the famous Poincaré statement “Suppose that in one night all the dimensions of the universe became a thousand times larger”, which is perfectly meaningful according to Poincaré, even if “The most exact measures will be incapable of revealing anything of this tremendous change, since the yard-measures I shall use will have varied in exactly the same proportions as the objects I shall attempt to measure” [15].
Here 0 and 1 are understood as classical truth values that are defined, for every w∈W, on a subset ψ R w of ψ R . Of course, σ w can be extended to ψ R in many ways. In particular, σ w could be seen as the restriction to ψ R w of a many-valued function, defined on ψ R and assigning truth values in the interval [0,1]. Such values could be interpreted as degrees of truth, or as probabilities of obtaining positive results when performing measurements, according to standard views in fuzzy logics. These interpretations of σ w would fit in well with the physical interpretation of the rfs of \(\mathcal {L}_{G}^{P}\) as sentences stating properties of individual objects, according to which true actually means certainly true (degree of truth 1): see Sections 5 and 6, and [24, 25, 31]. We maintain, however, that neither the notion of degree of truth nor the foregoing notion of probability should be seen as notions of truth alternative to the classical notion. Indeed, they can be expressed in terms of the classical truth of higher logical level sentences about rfs of ψ R . In any case, our arguments in this paper show that the notion whose properties are formalized by QL is different from all the notions mentioned above (classical truth, degree of truth, probability), and should not be interpreted as an alternative notion of truth.
It is interesting to observe that the ESR model, if objective, can be considered as an extreme case of a modal interpretation in which the possibility of assigning a truth value to all sentences of the form E(x) is assured by the interpretation of quantum probabilities as conditional (Section 3).
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Garola, C. Interpreting Quantum Logic as a Pragmatic Structure. Int J Theor Phys 56, 3770–3782 (2017). https://doi.org/10.1007/s10773-017-3309-7
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DOI: https://doi.org/10.1007/s10773-017-3309-7