Quantum metaphysical indeterminacy and worldly incompleteness


An influential theory has it that metaphysical indeterminacy occurs just when reality can be made completely precise in multiple ways. That characterization is formulated by employing the modal apparatus of ersatz possible worlds. As quantum physics taught us, reality cannot be made completely precise. I meet the challenge by providing an alternative theory which preserves the use of ersatz worlds but rejects the precisificational view of metaphysical indeterminacy. The upshot of the proposed theory is that it is metaphysically indeterminate whether p just in case it is neither true nor false that p, and no terms in ‘p’ are semantically defective. In other words, metaphysical indeterminacy arises when the world cannot be adequately described by a complete set of sentences defined in a semantically nondefective language. Moreover, the present theory provides a reductive analysis of metaphysical indeterminacy, unlike its influential predecessor. Finally, I argue that any adequate logic of a language with an indeterminate subject matter is neither compositional nor bivalent.

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  1. 1.

    The dominant theory of semantic indeterminacy is the supervaluationism of Fine (1975).

  2. 2.

    The target of Evans’ argument is indeterminate identity. While his result does not per se rule out the possibility of metaphysical indeterminacy sans indeterminate identity, as a matter of fact it had the effect of putting metaphysical indeterminacy in a bad light overall.

  3. 3.

    The most developed version of the account is Barnes and Williams (2011); see also Barnes (2010), Williams (2008a, b). The version of BW’s account which I employ in the present discussion is based on Williams (2008b). Alternative accounts of metaphysical indeterminacy include Akiba (2000, 2004), Wilson (2013); see Bokulich (2014) for a discussion of quantum metaphysical indeterminacy.

  4. 4.

    Barnes and Williams (2011, p. 114). The classical discussion of ersatzism, linguistic and otherwise, is Lewis (1986, pp. 136–91).

  5. 5.

    Akiba (2015) argued that BW’s notion of metaphysical indeterminacy is a consequence of uncontroversial facts about indeterminacy, and therefore that there is nothing specifically metaphysical about it. I will come back to this issue in Sect. 2.

  6. 6.

    The thesis that quantum mechanics provides examples of metaphysical indeterminacy has been articulated in Bokulich (2014), Lowe (1994, 1999), Williams (2008a) and Wilson (2013).

  7. 7.

    There exist results—most notably Bell’s theorem and the Kochen–Specker theorem—that impose strict constraints on any empirically adequate hidden variable theory. It is a consequence of the Kochen–Specker theorem that there is no hidden variable formulation of quantum mechanics, as long as the value of an observable is independent of how that value is measured. Bell’s theorem states that any hidden variable theory must be non-local, allowing some sort of action at a distance.

  8. 8.

    A classical reference on infinitary logic is Dickmann (1975).

  9. 9.

    Cf. Putnam (1979, p. 185): “A system has no complete description in quantum mechanics; such a thing is a logical impossibility”.

  10. 10.

    I am here ignoring the fact that E must include general information about w, such as the quantum-mechanical laws, as well as particular information about the intrinsic properties of e, and some necessary truths.

  11. 11.

    “Any Hermitian operator on a given space will invariably be associated with some measurable property of the physical system connected with that space...Any vector whatever in a given space will invariably be an eigenvector of some complete Hermitian operator on that space. That...will entail that any quantum state whatever of a given physical system will invariably be associated with some definite value of some measurable property of that system.” Albert (2009, p. 41).

  12. 12.

    One may rejoin that my reply conflates Hermitian operators with properties. For, goes the objection, the fact that every quantum property is captured by some Hermitian operator by no means entails that every Hermitian operator captures some quantum property. Consequently, I have failed to show that every state of a system is sharp with respect to some property. (I owe this observation to an anonymous referee.) The objection can be dealt with by appealing to the standard distinction between an abundant and a sparse conception of properties. When the objector raises doubts about the claim that every Hermitian operator captures some quantum property, what she has in mind is arguably the sparse conception of a physical property, or observable. The idea is that, even though one can define infinitely many Hermitian operators in a Hilbert space, it is highly doubtful that to each there corresponds some genuine physical property. But of course I do not need to endorse such a preposterous view. The notion of property, or observable, that I presuppose in my reply to the objection from expressive incompleteness is the abundant one, according to which a property is the semantic value of a condition definable in the relevant language. In the present case the language is the language of quantum mechanics, and the semantic values of the conditions definable in that language are Hermitian operators. Therefore, there is no risk of running out of properties in the relevant sense.

  13. 13.

    Cf. Skow (2010, p. 858), who considers and rejects a modification of the BW account in which the ersatz worlds are defined in “a language which suffers from semantic indeterminacy”.

  14. 14.

    On the topic of free logics and their semantics, see Nolt (2006).

  15. 15.

    Incidentally, that thesis is validated by the quantum logic of Birkhoff and von Neumann (1936), where the behavior of sentential connectives is read off of the structure of Hilbert spaces. It is worth remarking, however, that my argument from the previous section concerning the truth value of \(\bigvee _{i\in I}\phi (e,x_{i})\) is independent of the acceptance of quantum logic.

  16. 16.

    In BW’s theory a sentence is valid just in case it is a member of all precisificational possibilities for all worlds. Since BW’s ersatz worlds are classical maximally consistent sets of sentences, every classical tautology is valid. On the other hand, indeterminate sentences are neither true nor false. Thus, BW’s logic is neither bivalent nor compositional. An important caveat: not all versions of BW’s account have those features. In particular, the logic of indeterminacy developed in Barnes and Williams (2011) is both bivalent and compositional.

  17. 17.

    Of course, \(\exists z\phi (e,z)\) and \(\bigvee _{i\in I}\phi (e,x_{i})\) are not logically equivalent if the class of position values is contingent. But that issue is orthogonal to the problem being discussed. Thus, to avoid pointless complications, I am setting the issue aside.

  18. 18.

    I also have more general misgivings about Wilson’s theory of metaphysical indeterminacy. First of all, it is unable to model metaphysically indeterminate existence (cf. Barnes and Cameron 2009), because it would require existence and nonexistence to be determinates of some determinable—an implausible view. Second, I think that none of the examples of glutty metaphysical indeterminacy offered in Wilson (2013) are adequately motivated (cf. Bokulich 2014). Finally, Wilson regards open future claims as expressing instances of metaphysical indeterminacy. Although I agree with her on that, I doubt that indeterminacy about the open future can be given a determinable-based account (cf. Barnes and Cameron (2009). A comprehensive assessment of Wilson’s theory of metaphysical indeterminacy goes beyond the scope of this paper.


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I would like to express my gratitude to Axel Barceló, Aldo Filomeno, Eduardo García-Ramírez, John Horden, Ricardo Mena, Elias Okon and Jessica Wilson for helpful and constructive feedback on previous versions of this paper. I also would like to thank the audience of the workshops Logical Space. Logical and metaphysical issues and Philosophical Aspects of Modality, which took place at the Instituto de Investigaciones Filosóficas, UNAM, in September 2016. This work was supported by the PAPIIT Grant IA400316.

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Torza, A. Quantum metaphysical indeterminacy and worldly incompleteness. Synthese (2017). https://doi.org/10.1007/s11229-017-1581-y

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  • Ersatz world
  • Ersatzism
  • Superposition
  • Quantum mechanics
  • Quantum physics
  • Nonclassical logic
  • Compositionality
  • Bivalence
  • Semantic completeness
  • Supervaluationism
  • Metaphysical indeterminacy
  • Possible worlds
  • Impossible worlds