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ABSTRACT
It has been argued that quantum mechanics forces us to accept the existence of metaphysical, mind-independent indeterminacy. In this paper, we provide an interpretation of the indeterminacy involved in the quantum phenomena in terms of a view that we call Plural Metaphysical Supervaluationism. According to it, quantum indeterminacy is captured in terms of an irreducibly plural relation between the actual world and various misrepresentations of it.
Acknowledgements
We would like to thank two anonymous referees for their comments and suggestions. We also thank Claudio Calosi, Jonathan Schaffer, Jonathan Tallant, Alessandro Torza, the participants of a session of the eidos-seminar at the University of Geneva, the audience at the 4th Society for the Metaphysics of Science Conference, and the participants of the Quantum Indeterminacy Workshop at Dartmouth University, for useful comments on previous versions of this manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 The debate originates with Evans (Citation1978). See e.g. Akiba and Abasnezhad (Citation2014). See also Rosen and Smith (Citation2004) and Barnes (Citation2010). The received view is well exemplified by Lewis’s claim that ‘[t]he only intelligible account of vagueness locates it in our thought and language’ (Citation1986, 212).
2 Heisenberg has often suggested a similar view (see the collection of papers from Citation2007). Schrödinger (Citation1935) considers this option (the fifth section of the ‘cat-experiment’ paper is entitled ‘Are the variables really blurred?’), but eventually rejects it.
3 Bokulich (Citation2014, 460), for instance, is explicit that an understanding of QI might provide good grounds for interpreting orthodox QM in a realist fashion.
4 See in particular (Barnes Citation2010; Barnes and Williams Citation2011; Barnes Citation2013; Williams Citation2008). We take the version developed in Barnes and Williams Citation2011 to be the definite version of their theory.
5 This feature of vague predicates renders them susceptible to the sorites paradox, to which supervaluationism offers a solution. See (Fine Citation1975; Keefe Citation2000, §7-8) for influential defenses of the theory.
6 Note that we will later qualify the notion of precisification by distinguishing between the relations of admissibility, which holds between a possible universe and a model, and of co-precisification, which holds between ersatz worlds in a model and which corresponds to the accessibility relation in standard modal logic. See §4.2.1.
7 We use double lowercase letters to indicate that a constant or variable is plural, as is standard in plural logic. See e.g. Oliver and Smiley (Citation2016).
8 Contrast this to a case in which the protesters form a less tight chain, so that one or two of them could leave without breaking it.
9 See (Oliver and Smiley Citation2016, 212).
10 Note that we will discuss the aptness of this label for Rpp in light of its role with respect to the model theory in §4.2.1.
11 The notion of possibility here and in the remainder of the paragraph must be wider than physical/nomic possibility in the sense that it allows for completely precise ersatz worlds to count as possible precisifications of reality. Logical possibility fits this bill, but so might metaphysical possibility.
12 Note that ‘state of affairs’ is here used for illustrative purposes only and is not meant to indicate that our theory presupposes an ontology of states of affairs.
13 As we will see, in the case of the QI, none of those logically possible states of affairs can nomologically obtain.
14 We are grateful to an anonymous reviewer for helping us clarify the analogy and the following explanation.
15 We rely on the context to distinguish between use and mention throughout this section and the whole paper.
16 Note that this definition does not amount to a reductive analysis of indeterminacy; like B&W’s theory, our theory is non-reductive, as indicated in section 2.1.
17 Note that this means that our semantics corresponds to a restricted variant of Fine’s supertruth-based semantics developed in Fine (Citation1975) which is limited to complete models and in which models correspond to possible worlds. It closely follows the semantics for semantic supervaluationism due to Asher, Dever, and Pappas (Citation2009).
18 Cf. for example Chalmers (Citation2006), especially §3.
19 I.e. where φ,ψ are formulas, Δ is a set of formulas, and ⊨S is the supertruth-based notion of logical consequence, it is not generally the case that if Δ∪{φ}⊨Sψ, then Δ⊨Sφ→ψ. See Fine (Citation1975, 290) and Asher, Dever, and Pappas (Citation2009, 909). Counterexample: let Δ be empty and substitute p for φ and Dp for ψ. With ⊨S expressing conservation of supertruth, i.e. conservation of truth in all worlds in all supervaluationist models, we have that if ⊨Sp, then ⊨SDp, but not that ⊨Sp→Dp, since there are models containing worlds in which p is true, but not Dp.
20 I.e. we have that if Δ∪{φ}⊨ψ, then Δ⊨φ→ψ. The counterexample described in note 16 does not arise, since p⊨Dp requires the conditional p→Dp to be true in all worlds in all models, which just means that ⊨p→Dp.
21 See Asher, Dever, and Pappas (Citation2009, 922–923). This feature of the logic is incidentally very much in line with the way cases of QI are handled in PMS, since in these cases, the determinate truths are imposed on the intended model by the physics. See §5.3.
22 But need not. As we have mentioned in §4.2.2, the semantics we present here meets the classicality desideratum formulated by Barnes and Williams and Calosi and Wilson. If one rejects this desideratum, one can instead develop a version of PMS which incorporates a non-classical logic and semantics.
23 KS establishes a contradiction for a Hilbert space of 3 dimensions with 117 vectors, and this is why the proof is particularly complicated. A simpler proof is in Cabello, Estebaranz, and Garcìa-Alcaine (Citation1996), where the Hilbert space is 4d, with 18 vectors. Cabello’s proof is however weaker, because any contradiction in 3d is also a contradiction in 4d while the converse is not true.
24 Some interpretations of QM maintain that this is the case at the price of considering observables like spin as contextual (i.e. dependent on the measurement context). Since non-contextuality is an assumption of KS, by dropping we avoid value indefiniteness. That is why, to be precise, what the theorems establish is the impossibility of non-contextual hidden-variables theories.
25 They adopt this notion from situation semantics. See e.g. Barwise and Perry (Citation1983).
26 See Corti (Citationforthcoming) for a critical discussion of Darby & Pickup’s proposal.
27 Those claims would need some fine honing, given that variant domain versions of possible world semantics and the possibility of alien properties, but the point we are making here does not require such a level of refinement. Capturing the difference between global possible worlds and local situations would still require the specification of what situations are abstracted from.
28 A natural question to ask is where we find the laws in PMS. A way to address this issue is by accepting a bifurcated treatment of laws of nature, which draws on a well-established distinction between two kinds of physical laws, namely the dynamical and kinematical ones. Maudlin (Citation2017) for instance, speaks of FLOTEs—Fundamental Laws of Temporal Evolution—to refer to the former, and adjunct principles to refer to the latter. We could suppose that the only laws that need to be recovered are the kinematical ones, since these are the laws that generate indeterminacy (namely, those that we cannot have as sentences within the ersatz worlds). The kinematics could potentially be recovered at the meta-level as a set of constraints across the possibilities; after all, this is precisely what the kinematics is. Such a view, though admittedly underdeveloped, does not seem to conflict irremediably with our understanding of physical laws. Be that as it may, we stress that this issue is independent from whether or not the ersatz worlds must contain all the laws as propositions in PMS; as we argued, there is no reason why they should.
29 See e.g. Albert and Loewer (Citation1988). Thanks to Jonathan Schaffer here.