Quantum metaphysical indeterminacy


On a wide variety of presently live interpretations, quantum mechanics violates the classical supposition of ‘value definiteness’, according to which the properties (‘observables’) of a given particle or system have precise values at all times. Here we consider whether two recent approaches to metaphysical indeterminacy—a metaphysical supervaluationist account, on the one hand, and a determinable-based account, on the other—can provide an intelligible basis for quantum metaphysical indeterminacy (QMI), understood as involving quantum value indefiniteness. After identifying three sources of such QMI, we show that previous arguments (Darby in Australas J Philos 88:227–245, 2010; Skow in Philos Q 60:851–858, 2010) according to which supervaluationism cannot accommodate QMI are unsuccessful; we then provide more comprehensive arguments for this conclusion, which moreover establish that the problems for supervaluationism extend far beyond the orthodox interpretation. We go on to argue that a determinable-based approach can accommodate the full range of sources of QMI.

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  1. 1.

    It has also been suggested that quantum objects may be metaphysically indeterminate in failing to have determinate identity; see, e.g., Lowe (1994) and French and Krause (2003). Here we focus only on property value indeterminacy.

  2. 2.

    Other accounts aiming to characterize MI include those in Smith and Rosen (2004) and Torza (2017); beyond some notes, we leave discussion of these alternatives for another day.

  3. 3.

    See Wilson (2016) for reasons to reject taking MI to generate propositional indeterminacy.

  4. 4.

    There are, of course, many understandings (readings, versions) of each ‘interpretation’, appealing, e.g., to different underlying ontologies. Though we can’t consider the prospects for supervaluationism vis-á-vis every variation on a given interpretational theme, for dialectical purposes it will suffice to show that the problems for supervaluationism do in fact attach to the orthodox interpretation and moreover extend to common understandings of several live non-orthodox interpretations.

  5. 5.

    Wallace (ms.) claims that one might resist the supposition that even the orthodox interpretation is committed to EEL, on grounds that the practice of physicists doesn’t rely on it. Wallace clarifies that the considerations he raises aren’t aimed at showing that the best interpretation of quantum mechanics is one which abandons EEL, but in any case attention to the considerations he raises must await another occasion; meanwhile we continue on under the assumption that, as above, indications of QMI commonly proceed by way of EEL.

  6. 6.

    In what follows, we use the same notation for both observables and operators; strictly speaking, operators are mathematical objects representing observables.

  7. 7.

    Entanglement might also be present in a simple system (see Hasegawa 2012), in which case the characterization of entanglement MI would need tweaking.

  8. 8.

    Again, we advert to EEL as reflecting a convenient and “fairly standard” way of thinking about when quantum observables have determinate values; but when relevant will discuss the bearing of other linking principles.

  9. 9.

    See Darby (2010, pp. 237–238), whose presentation follows Hughes (1989), and Skow (2010, pp. 854–856). For a detailed introduction, see Held (2018).

  10. 10.

    See, e.g., Peres (2002).

  11. 11.

    Thanks to Nina Emery for this point. Note also that this sort of restriction on which quantum theories or associated possibilities are relevant to assessing the adequacy of a given account of QMI isn’t in tension with the goal of offering a metaphysical account of such indeterminacy, contrasting with semantic or epistemic accounts of such indeterminacy.

  12. 12.

    Torza (2017) offers an account of MI which explicitly abandons complete precisifications; consideration of this account and of whether it is a version of the supervaluationist strategy under consideration is a substantive question, which as noted we leave for another day.

  13. 13.

    Thanks to Michael Miller here. To be sure, there are other difficulties with the partial precisification strategy, which we will highlight down the line. Our present point is simply that there are available supervaluationist responses to Skow’s specific concern with this strategy.

  14. 14.

    This response was suggested by Ross Cameron (p.c.).

  15. 15.

    Thanks to an anonymous referee for suggesting this motivation.

  16. 16.

    For discussion of why the tails are necessary, see Albert and Loewer (1992).

  17. 17.

    Thanks to Nina Emery and Heather Demarest here. As Emery observed in her AOC comments: “[the] fuzzy link will give rise to MI in the paradigm case in just the same way as [...] the eigenstate-eigenvalue link”.

  18. 18.

    Hence in describing GRW with flashes, Esfeld (2014) says, “The flashes are all there is in space-time. That is to say, apart from when it spontaneously localizes, the temporal development of the wave-function in configuration space does not represent the distribution of matter in physical space. It represents the objective probabilities for the occurrence of further flashes, given an initial configuration of flashes. As in [Bohmian Mechanics], there hence are no superpositions of anything existing in physical space” (100).

  19. 19.

    As Esfeld (2014) notes, on this approach, macroscopic objects are “well localized” (100) but not determinately so.

  20. 20.

    The text in brackets has been altered for compatibility with our discussion.

  21. 21.

    Bokulich (2014, p. 465) uses a similar case to illustrate why position-momentum MI cannot be given an epistemic interpretation.

  22. 22.

    Hence: “In the Bohmian mechanical version of nonrelativistic quantum theory, quantum mechanics is fundamentally about the behavior of particles; the particles are described by their positions, and Bohmian mechanics prescribes how these change with time” (Goldstein 2017, p. 13).

  23. 23.

    Considerations of space prevent our discussing modal interpretations, on which only a proper subset of quantum properties are taken to have definite values, but in our view supervaluationism will also run afoul of such interpretations, due to a modal variant of the Kochen-Specker-theorem (see Domenech et al. 1981). We plan to discuss modal interpretations in more detail in future work.

  24. 24.

    Torza (2017) moreover argues that implementing the partial precisification strategy (a version of which he endorses) requires rejecting classical compositional semantics. Note that there is no associated difficulty for a determinable-based object-level account: on this account MI does not generate any propositional indeterminacy, so compatibility with classical logic and semantics does not need to be regained, so to speak, by appeal to what is true in all ‘maximal and classical’ precisificationally possible worlds.

  25. 25.

    It may be worth noting that these considerations bear negatively on Robbie Williams’ suggestion (p.c.) that the supervaluationist could avoid running afoul of the Kochen–Specker theorem by singling out a proper subset of observables as physically relevant, and appealing to partial precisifications where only these observables were determinate. Implementing the suggestion would require that, for every set of incompatible observables, the supervaluationist identify only one as physically relevant; but in general—notably, for components of spin—there will be no plausible non-arbitrary way of doing this. To be sure, there remains the radical strategy of taking only one quantum observable to be physically relevant, which would indeed sidestep our arguments, since (as prefigured) these rely on there being certain dependencies between different (physically relevant) quantum observables. But in removing dependencies between real quantum observables the supervaluationist throws the baby out with the bathwater, for any QMI there may be stems from these dependencies. Hence it is that the salient interpretation on which there is a sole privileged observable—namely, Bohmian mechanics—is one on which the observable is always determinately-valued, and any seeming indeterminacy is merely epistemic.

  26. 26.

    Compare an endurantist (or ‘three-dimensionalist’) position as regards the persistence of objects, according to which bare claims of the form ‘Object O has property P’ are not truth evaluable, since incomplete; rather, what is truth-evaluable are (on one salient strategy; see Haslanger (1989) for discussion) claims of the form ‘Object O has property P at time t’.

  27. 27.

    See especially note 30.

  28. 28.

    Here we agree with Wolff (2015) (and also with an anonymous referee) that an appeal to perspectives or measurement contexts invites an (incorrect) reading of the multiple determination at issue according to which “different determinates of the same determinable [...] could be instantiated at the same time, but would need to be ‘looked at’ from different perspectives” (384).

  29. 29.

    It may be worth observing that the present suggestion can be understood as a variation on a many-worlds approach to macro-superpositions, where the relative states are (a) instantiated within a single world, and (b) apply to micro-superpositions as well as macro-superpositions.

  30. 30.

    On the suggestion at hand, it is unclear that it makes sense to apply a degree-theoretic approach to gappy QMI, at least for cases where no determinates of the determinate are instantiated, on pain of violating certain quantum constraints; hence we do not pursue this strategy here.

  31. 31.

    To be clear: this description makes sense on the supposition that the superposition is not collapsed via measurement or otherwise resolved. It is important to register that on a degree-theoretic understanding of (e.g.,) superposition-based QMI of the sort operative in the double-slit experiment, the claim is not that, insofar as the particle instantiates both determinates to a certain specific degree (\(< 1\)) when in a superposition state, one will thereby be in position to measure the particle as passing through a given slit to the associated degree. Such a result would violate well-known empirical results, but in any case on the degree-theoretic approach under consideration here, the operative assumption is that, as per usual, measurements resolve the superposition in such a way that one only ever measures the particle as having gone through one of the slits (that is, as instantiating the relevant determinate to degree 1). Thanks to Michael Miller for discussion here.

  32. 32.

    It is also true that on glutty implementations, the object having the determinable property “fails to have a definite value” of the determinable—at least, simpliciter, or to degree 1. In context, Bokulich is clear that she has in mind a gappy implementation.

  33. 33.

    Torza (2017) objects that a gappy implementation violates the supposition that, e.g., claims that a system has a determinable position should be formalized in existential terms as the having of some determinate position; but we reject this supposition as building in the reducibility of determinables to determinates.

  34. 34.

    This proposal suggests a reading of the quantum state in thoroughly relational terms, bearing some similarity to the relational interpretation advanced in Rovelli (1996) and discussed in Laudisa and Carlo (2013).


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Thanks to audience members at talks on this topic given at the ‘Metaphysics of Quantity’ conference (NYU, 2015), the Arizona Ontology Conference (2016), the Jowett Society (Oxford, 2016), an eidos seminar in Metaphysics (University of Geneva, 2016), the ‘Kinds of Indeterminacy’ workshop (University of Geneva, 2016), and the Philosophy of Physics seminar at the Instituto de Investigaciones Filosóficos (UNAM, 2018). Thanks also to Nina Emery for her excellent AOC comments, and to Alisa Bokulich, Eddy Chen, Heather Demarest, Catharine Diehl, Benj Hellie, Michael Miller, Alyssa Ney, Elias Okon, Alessandro Torza, Johanna Wolff, and students in Wilson’s ‘Varieties of Indeterminacy’ seminar (2018) for helpful discussion. Finally, thanks to the Swiss National Science Foundation (Project Numbers BSCGIo_157792 and 100012_165738) and the Social Sciences and Humanities Research Council for funding support.

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Calosi, C., Wilson, J. Quantum metaphysical indeterminacy. Philos Stud 176, 2599–2627 (2019). https://doi.org/10.1007/s11098-018-1143-2

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  • Quantum mechanics
  • Metaphysical indeterminacy
  • Quantum indeterminacy
  • Metaphysical supervaluationism
  • Determinable-based metaphysical indeterminacy