Yet again, quantum indeterminacy is not worldly indecision

Abstract

It has been argued that non-relativistic quantum mechanics is the best hunting ground for genuine examples of metaphysical indeterminacy. Approaches to metaphysical indeterminacy can be divided into two families: meta-level and object-level accounts. It has been shown (Darby in Australasian Journal of Philosophy 88(2):27–245, 2010. https://doi.org/10.1080/00048400903097786; Skow in Philosophical Quarterly 60(241):851–858, 2010) that the most popular version of the meta-level accounts, namely the metaphysical supervaluationism proposed by Barnes and Williams (Oxford Studies in Metaphysics, Oxford University Press, Oxford, pp 103–148, 2011), fails to deal with quantum indeterminacy. Such a fact has been taken by many as a challenge to adapt supervaluationism to quantum cases. In this paper, I will focus on the very last of these attempts, i.e. the situation semantics account proposed by Darby and Pickup (Synthese 1–26, 2019). After having shown where quantum indeterminacy arises and having surveyed the assumptions endorsed by the participants of the debate, I turn to Darby and Pickup’s proposal. I argue that, despite the machinery introduced, their account still fails to account for quantum indeterminacy. After considering some possible counterarguments, I suggest in the conclusion that one can plausibly extend the argument to those meta-level approaches that treat quantum indeterminacy as worldly indecision.

1 Introduction

Metaphysical indeterminacy (henceforth ‘MI’) is the idea that indeterminacy is a feature of the world itself, rather than a feature imputable to the words we use (Fine 1975) or to our knowledge (Williamson 1994). MI has often been regarded as a priori incoherent. Such an attitude has been encouraged by three main facts. First, many thought that MI necessarily allows the possibility of indeterminate identities. Given the strength of Evan’s (1978) argument against such a position, the majority of philosophers have concluded that MI is impossible. Second, many’s attitudes on MI were influenced by authoritative philosophers like Russell (1923) and Lewis (1986), who explicitly banned MI as conceptually contradictory or, if coherent, blatantly impossible. Third, there were no positive reasons for thinking that MI was conceivable in the first place. Indeed, precise and convincing accounts of MI were nowhere to be found, leading many to the conclusion that MI is inconceivable.

The attitude toward the possibility of MI now seems to be more relaxed than before, especially because logically clear accounts of MI have been put forward in the literature. Moreover, many authors have argued that these accounts help make sense of different phenomena.¹

As a matter of fact, many think that the most compelling reason for taking MI seriously is that some interpretations of non-relativistic quantum mechanics, plus some extra assumptions, seem to suggest a genuine case of it. Even if the first examples of quantum MI proposed² are no longer seen as genuine, that the properties of quantum objects are metaphysically indeterminate is still a live option (Bokulich 2014; Wolff 2015; Calosi and Wilson 2018).

Following Wilson (2013), one may divide the accounts of MI available in the literature into two broad categories:³

• Meta-level accounts: usually, these approaches consider MI as worldly ‘indecision’ between different possible, mutually incompatible, precise states of affairs. In other terms, p is metaphysically indeterminate iff (i) there are at least two possible descriptions of a state of affairs involving p, i.e. p and $\lnot p$, and (ii) these possible descriptions that do not contain in themselves any semantic nor epistemic indeterminacy fail in both determinately representing and misrepresenting the actual world. The details vary slightly across models, but the general idea remains the same: a state of affairs is metaphysically indeterminate when the world itself is unsettled between (at least) two possible determinate states. Examples of meta-level accounts can be found in Akiba (2004), Barnes and Williams (2011) and Darby et al. (2017).

• Object-level accounts: this family of accounts claims that there are three possible states of affairs: the state that p, that $\lnot$p and the state that it is indeterminate whether p. MI is thus considered as a third state of affairs that cannot be reduced nor be understood in terms of the two definite states p and $\lnot$p. Arguably, Tye (1990) is one of the progenitors of this approach to MI, albeit his account seems not so popular anymore for different reasons.⁴ Other object-level accounts of MI can be found in Smith and Rosen (2004) and Wilson (2013).

The contemporary research in MI seems to be oriented toward the former family of accounts rather than the latter.⁵ In what follows, I focus on meta-level treatments of indeterminacy and how they deal with examples taken from non-relativistic quantum mechanics. In particular, I focus on the account recently developed in Darby et al. (2017) and Darby and Pickup (2019). I provide background on where and under what assumptions quantum indeterminacy is supposed to arise in Sect. 2, while Sect. 3 is devoted to a presentation of Darby and Pickup’s account.⁶ Then, I argue in Sect. 4 that this way of modelling metaphysical indeterminacy fails to account for quantum phenomena, even granting all the assumptions requested by Darby and Pickup’s view. After considering possible replies to the argument here presented (Sect. 5), I draw what I take to be the moral of all of this, namely, that quantum superposition cannot be understood as worldly indecision between definite states of affairs (Sect. 6).

2 Where quantum indeterminacy lies

That quantum mechanics offers a genuine example of MI is debatable. Indeed, such a claim relies on five main assumptions which are, by themselves, highly controversial. In this section, I briefly discuss them in turns.

First, the relevant literature concerning quantum MI assumes as a background an ‘orthodox’ interpretation of quantum mechanics.⁷ It must be acknowledged that some of the main players of the debate have argued that MI also arises in other interpretations⁸ as well; still, no complete and satisfying investigation in this sense has been put forward yet. Since orthodox quantum mechanics is seen as an instrumental view, the second assumption held is that being realist or antirealist is a stance one assumes toward a theory, rather than something embedded in the theory itself (following McMullin 1984).

The third assumption is that Hermitian operators correspond to properties instantiated by quantum systems;⁹ in particular, the relation between the properties instantiated and their mathematical description is captured by the ‘Eigenstate-Eigenvalue link’ (EEL):¹⁰

• (EEL) A quantum system has a definite value $v$ for an observable $O$ iff it is in an eigenstate of ${\hat{O}}$ having eigenvalue $v$.

This principle bridges the mathematical and the metaphysical descriptions. Under the assumption that the properties of a system (i.e. its observables) are modelled in quantum formalism by Hermitian operators, (EEL) gives us the sufficient and necessary conditions under which we can claim that a quantum system instantiates some properties: if a property O is represented by a Hermitian operator ${\hat{O}}$, the system instantiates that property if and only if its state is an eigenstate of ${\hat{O}}$.

There are many cases in which a system fails to be in an eigenstate of some operator. Let us consider spin-$\frac{1}{2}$ along different directions. Given some properties of the mathematical vector space in which spin properties are represented, if a system is in a definite state of, say, spin up along the x-axis ($|\uparrow \rangle _{x}$), it is necessarily in a superposed state of spin along the z-axis:

$$\begin{aligned} |\psi \rangle =|\uparrow \rangle _{x}=\frac{1}{\sqrt{2}}(|\uparrow \rangle _{z} + |\downarrow \rangle _{z}) \end{aligned}$$

(1)

Since $\frac{1}{\sqrt{2}}(|\uparrow \rangle _{z} + |\downarrow \rangle _{z})$ is not an eigenvector of the operator $\sigma _{z}$ (i.e. the operator correspondent to the property ‘having spin along the z-axis’), when the system is in $|\psi \rangle$, it does not have a definite value of spin along the z-axis. Friends of quantum indeterminacy claim that when a system is in a state like $|\psi \rangle$, or more generally, in a state that is not an eigenvector of some Hermitian operator, the system indeterminately instantiates the property that corresponds to that operator. These kind of states are (allegedly) responsible for the arise of quantum indeterminacy: when a quantum system is in one of these states, the instantiation of one of its properties might be indeterminate.

The fourth assumption is that such indeterminacy is metaphysical rather than semantic or epistemic. It is not semantic because equations are sharp and precise, and none of the imprecision connected with the ambiguities of natural languages is involved.¹¹ It is not epistemic because some mathematical theorems¹² show formally that if you try to cheat, so to speak, by assigning a definite value to all the properties, you contradict the empirically tested predictions of quantum mechanics. The general conclusion is then that such indeterminacy must be metaphysical.

Note finally that (EEL) per se is not enough for claiming that the system indeterminately instantiates some properties. As Glick (2017) argues, one may take the (EEL) as saying that when the state of a system is not an eigenvector of an operator, it lacks the corresponding property altogether. Then, the fifth assumption taken by the participants of the debate is what I dub ‘Property Nonetheless’ (PN):

• (PN) A quantum system always instantiates its (possible) observables (i.e. those properties that can be measured and represented mathematically through a Hermitian operator).¹³

This principle should guarantee that a system always instantiates its properties, making room for the possibility of indeterminacy. Indeed, (PN) and (EEL) jointly implies that when the system is not in an eigenstate of one of its observables, it does not possess a definite value for the corresponding properties. The fact that these values are not definite allows us to interpret it as if the system indeterminately instantiates the property mentioned above.

Most of the assumptions above are controversial. The (EEL) has been strongly criticized in different interpretations of quantum mechanics (Wallace 2019; Albert and Loewer 1996); the claim that indeterminacy is not semantic nor epistemic might be resisted (either assuming an empiricist stance toward science or accepting some $\psi$-epistemic interpretation of quantum mechanics). Insofar as, in general, issues on quantum mechanics tend to be highly controversial, such a list could continue forever. Since these assumptions are necessary for arguing that quantum mechanics leaves open the possibility of MI, reasonably, they are also endorsed by Darby and Pickup. So far as the paper aims at showing that Darby and Pickup’s account fails to account for quantum indeterminacy I uncritically endorse, for the sake of argument, the assumptions above in what follows.

Having considered the assumptions needed in order to claim that quantum mechanics offers a genuine example of MI, we now turn to how meta-level approaches deal with quantum MI.

3 Meta-level accounts of quantum indeterminacy

Barnes and Williams (2011) proposed a supervaluationist account of MI. According to them, when there is MI, there are at least two ersatz possible worlds (that do not contain any semantic or epistemic indeterminacy in themselves) such that they both fail to represent reality correctly. However, these possible worlds do not determinately misrepresent reality either. So, if it is metaphysically indeterminate if p, then there is a possible world in which p obtains and a possible world in which $\lnot p$ obtains, but it is in principle¹⁴ impossible to say which of these represent the concrete actual world. It has been argued that their proposal fails to account for the indeterminacy that arises in quantum mechanics: as Darby (2010) and Skow (2010) independently showed, the Kochen–Specker theorem implies that there is at least one case in which it is impossible to assign simultaneously (in a context-independent way) a definite value to all the properties of a quantum system. In Barnes and Williams’ account, the ersatz possible worlds are entirely defined and determinate; that is, in every possible world, all the properties of every system always have a definite value. Since the Kochen–Specker theorem shows that there is at least one quantum system such that it is impossible to assign definite values to all its properties, no possible worlds (of the kind employed in Barnes and Williams’ proposal) may be used to account for the indeterminacy of this quantum system. Since there is at least a quantum system whose indeterminacy cannot be accounted for using Barnes and Williams’ view, their account of MI is not apt for explaining quantum indeterminacy in general.

The failure of Barnes and Williams’ account has been taken by many as a challenge to develop a supervaluationist treatment of MI able to deal with quantum cases.¹⁵ In what follows, I focus only on the most recent attempt to develop a supervaluationist account of MI, namely the view proposed by Darby and Pickup (2017, 2019).

The account proposed by Darby and Pickup consists of using a situation semantics instead of the machinery of possible worlds. The core idea of situations semantic is that the truth-value of a proposition¹⁶ has to be assessed relative to a situation; where situations are parts of our concrete actual world, i.e. situations are non-representational concrete truth-makers. In the situation semantics framework, propositions are not true or false simpliciter; rather, their truth-value is relative to the situation at which they are evaluated.¹⁷ Hence, truth (and falsehood) simpliciter is replaced with ‘truth (falsehood) at a situation.’

Insofar as situations are intrinsically partial, they do not tell everything about a certain state of affairs. In those situations that lack an aspect of reality, the proposition expressing that part is taken by Darby and Pickup as lacking a truth-value.¹⁸ An example will illustrate how situation semantics works. Take a statement like ‘Alice is playing with her dog’. Such a proposition is true if Alice is playing with her dog in the relevant situation and it is false if she is jogging (or doing something different than playing with her dog) in the relevant situation. In those situations in which there is no Alice, like in the situation in which there is a ski race on a mountain distant from Alice, the proposition ‘Alice is playing with her dog’ lacks a truth-value. Following Barnes and Williams, Darby and Pickup try to model indeterminacy as a matter of truth-values in situations which are candidates for representing actuality. The criteria of ‘candidates for representing actuality’ employed seems to be the following: a situation is a candidate for representing actuality if it does not determinately misrepresent¹⁹ the actual world. In the previous example, if Alice is playing with her dog in the actual world, then any situation in which the proposition ‘Alice is playing with her dog’ is true is a candidate for representing actuality; all the situations that determinately misrepresent the actual world, i.e. those in which Alice is doing something else, are not candidates for representing actuality. Given the fact that they accept that propositions can lack a truth-value at a situation, also situations in which there is no Alice will be candidates for representing actuality; obviously, this is true only if these situations do not misrepresent the actual world in other regards. For example, if in the actual world, while Alice is playing with her dog, there is a ski race on some distant mountain, then both the situation with Alice playing and the situation in which there is a ski race going on are candidates for reality.

Using the machinery of situation semantics, they propose to model MI as a genuine disagreement between those situations that are candidates for representing actuality. When a state of affairs is determinate, the relevant situations, i.e. those that are candidates for representing actuality that describe such a state of affairs, never disagree. In the previous example, for instance, the situation with Alice playing with her dog in some places does not contradict the situation with a ski race going on in some distant mountain. However, they claim that, if there is MI, then we have different situations that disagree on what reality is like, despite the fact that all of them are candidates for representing actuality.²⁰ To capture such a disagreement, they propose the following criteria:

Determinacy A proposition p is determinate iff it is true in some situation which is a candidate for representing reality and false in no such situation.

Indeterminacy A proposition p is indeterminate iff it is true in some situation which is a candidate for representing reality and false in some other such situation. (Darby and Pickup 2019, p. 9)

According to Darby and Pickup’s account, a proposition is determinately true when it is not false at any situation that is a candidate for representing actuality and true in at least one.²¹ In contrast, a proposition is indeterminate when there are at least two situations that are candidates for representing actuality, and the proposition is true at one of them and false at the other.

Again, an example helps us to understand better how the machinery works. Let us suppose that it is metaphysically indeterminate whether Alice is playing with her dog. In the set of situations which do not determinately misrepresent the actual world, there will be many situations that do not say anything about Alice; in respect of these situations, the proposition ‘Alice is playing with her dog’ simply lacks any truth-value. All the other situations will be of two kinds. Some situations which are candidates for representing reality make true that Alice is playing with her dog; however, the same proposition will be false at other situations which are candidates for representing actuality. Given that the situations in which the proposition is true are candidates for representing actuality as good as those in which the proposition is false, it is metaphysically indeterminate whether Alice is playing with her dog in the actual world.²²

Let us turn now to how they deal with quantum superposition. They consider the well-known example of Schrödinger’s cat: a cat is closed in a diabolic box that contains a device that releases a poisonous gas if an atom of isotope uranium decays. If the atom decays, the cat drops dead; if it does not decay, the cat survives. According to quantum mechanics, the system evolves in a way that, after a certain time, the atom will be in a superposition of being decayed and being not; since the atom and the cat compose a bigger system, the upshot of the thought experiment is that also the cat ends up in a superposition of being alive and dead. How can we make sense of such indeterminacy? This is Darby and Pickup’s proposal, which I quote at length:

Our model can deal with this case in a natural way. Consider the following three situations, each of which is a candidate for representing actuality: s$_{1}$, a situation in which the cat is alive, s$_{2}$, a situation in which the cat is dead, and s$_{3}$, the fusion of s$_{1}$ and s$_{2}$. In s$_{1}$, it is true that the cat is alive and false that it is dead. In s$_{2}$ it is true that the cat is dead and false that it is alive. So, what status do these propositions have? According to our definition, they are indeterminate, because there is a situation which is a candidate for representing actuality in which they are true and another situation which is a candidate for representing actuality in which they are false. What is the case in s$_{3}$? It seems s$_{3}$ is overdetermined with respect to the cat’s mortality, as parts of s$_{3}$ disagree about whether the cat is alive. Specifically, s$_{2}$ precludes the proposition that the cat is alive and s$_{1}$ precludes the proposition that the cat is dead. So, following our argument above, the proposition that the cat is alive is neither true nor false in s$_{3}$ (as is the proposition that the cat is dead). The indeterminacy involved in superposition is therefore dealt with. (2019, p. 16)

One of the crucial aspects of their account that emerges in the quotation is the following: if two contradictory propositions (e.g. p and $\lnot p$) are going to be verified in the same situation, then they both lack a truth-value at that situation. Such a detail is crucial for avoiding the Kochen–Specker result. Recall why Barnes and Williams’ account fails with quantum indeterminacy: since they use complete possible worlds that do not contain any indeterminacy in themselves, they must attribute definite values to all of a quantum system’s properties at the same time. Darby and Pickup’s account instead, successfully manages to avoid such a contradiction. Indeed, they argue that different situations describe different properties of the system but, crucially, no incompatible properties will receive a definite value in the same situation. Insofar as a situation might be silent on some aspect of reality, there is no need to assign definite values to all the properties of a quantum system. Using disagreeing situations, and avoiding any contradiction arising from assigning values to incompatible properties in the same situations, their machinery of situations accounts for quantum indeterminacy.

4 A situation in which situation semantics fails

Albeit prima facie Darby and Pickup’s account seems to accommodate cases of quantum indeterminacy, I argue now that it is not the case.

Let us start by assuming that in the actual world an electron is in the state:

$$\begin{aligned} |\psi \rangle =\frac{1}{\sqrt{2}}(|\uparrow \rangle _{z} + |\downarrow \rangle _{z}) \end{aligned}$$

(2)

Note that according to the formalism of quantum mechanics, state (2) is mathematically equivalent to:

$$\begin{aligned} |\psi \rangle =|\uparrow \rangle _{x} \end{aligned}$$

(3)

Indeed, being in a superposition on the z-axis means having a definite value for the x component of spin.²³ Note that this implies that being in spin up along the x-axis means being in a superposition of the z-axis; in other terms, $|\uparrow \rangle _{x}$ and $\frac{1}{\sqrt{2}}(|\uparrow \rangle _{z} + |\downarrow \rangle _{z}$ are mathematically equivalent ways of describing the same vector (i.e. the vector is represented in different bases of the Hilbert space).

Darby and Pickup’s account deals with superposition in the following way: when a system is superposed on an axis, there is a situation in which the system has spin up, and a situation at which it has spin down. Therefore, if in the real world our electron is in state $|\psi \rangle$, then there are at least two situations that are candidates for representing actuality:²⁴ at s$_{1}$ the particle has spin-up along the z-axis and at s$_{2}$ it has spin-down along the z-axis. Darby and Pickup propose to capture the two properties that the electron could have through a couple of propositions. The proposition expressing the fact that the electron has spin up is true at the first situation and false at the second; the opposite is true for the proposition expressing the property of having spin down along the z-axis. Albeit they do not specify what such a proposition is, I propose to consider the following:

($p_{z^{+}}$):

The system is in a state that is an eigenvector of the operator $\sigma _{z}$ with a corresponding eigenvalue +1.²⁵

This proposition, which captures the fact that the system has spin up along the z-axis, is true at s$_{1}$, and false at s$_{2}$ since it makes true a different proposition that entails the falsehood of ($p_{z^{+}}$).²⁶ Note that ($p_{z^{+}}$) is also false when the system is in a superposed state like (1).²⁷ Indeed, a superposed state of eigenvectors is not itself an eigenvector of its corresponding operator. That is to say that a superposition of the eigenvectors (with different eigenvalues) of the operator $\sigma _{z}$, is not itself an eigenvector of $\sigma _{z}$ (but crucially, it will be an eigenvector of some other operator; in the case at hand, of $\sigma _{x}$ when the expansion coefficients of the superposition are $\frac{1}{\sqrt{2}}$).

Let us focus now on their criteria of ‘candidate for representing actuality’. A situation is a candidate for representing actuality if it does not determinately misrepresent the actual world. An example is the situation s$_{3}$ that they explicitly accept, i.e. the situation that should verify both s$_{1}$ and s$_{2}$. In this situation, ($p_{z^{+}}$) and ($p_{z^{-}}$) both lack a truth value. However, given the definition of ‘candidate for representing actuality,’ we can find many other situations concerning our electron that Darby and Pickup do not consider. Take the proposition:

($p_{x^{+}}$):

The system is in a state that is an eigenvector of the operator $\sigma _{x}$ with a corresponding eigenvalue +1,

There will be a situation (s$_{4}$) candidate for actuality which makes true only ($p_{x^{+}}$).

At this stage of the argument, I am going to endorse the following principle, which I dub the principle of ‘equivalent candidates for representing actuality’ (ECA henceforth):

• (ECA) Consider a situation s$_{1}$ that is a candidate for representing actuality and verifies only a proposition $p_{1}$ which contains a mathematical object $o_{1}$. Any other situation s$_{2}$ that differs from s$_{1}$ only in that it makes true a proposition $p_{2}$ which is obtained by simply replacing $o_{1}$ with $o_{2}$, where the latter is a mathematical object equivalent to the former (i.e. $o_{1}=o_{2}$), is also a candidate for representing actuality.

Such a principle seems to be intuitively reasonable. Let us see how it works by presenting a toy example. Suppose it is metaphysically indeterminate how many oranges there are in the fridge; assume further that there might just be either three or four. According to Darby and Pickup’s view, there is a possible situation in which there are three oranges, and a possible situation in which there are four, but neither describes correctly nor misrepresents the actual world (and therefore the propositions ‘there are three oranges in the fridge’ and ‘there are four oranges in the fridge’ are indeterminate, being true and false in at least one situation). The principle (ECA) simply guarantees that if the possible situation in which there are three oranges is a candidate for representing actuality, then also the situations that verify respectively only the propositions ‘there are two plus one oranges in the fridge’ or ‘there are four minus one oranges in the fridge’, and so on, are candidates for representing actuality. All of this might seem superfluous, but it is not. In everyday examples like the one above, there will usually be a unique situation that makes true every mathematically equivalent proposition. However, it is better to assume that it is not the case for quantum phenomena. Darby and Pickup (2019, p. 11) explicitly say that a situation that verifies a proposition concerning, say, spin along the x-axis may not verify any proposition concerning spin along a different direction. To avoid begging the question against their view, I assume that there are different situations that make true (or false) propositions concerning spin in different directions.²⁸

Coming back to quantum indeterminacy, we have seen that the state spin up along the x-axis is mathematically equivalent to being in a superposition of spin up and down along the z-axis. That is, they are the very same state from a mathematical point of view.²⁹ The same can be said of the latter states, i.e. being in spin up and down along the z-axis means being in a superposition of spin up and down along the x-axis. To put it explicitly:

$$\begin{aligned} |\uparrow \rangle _{z}=\frac{1}{\sqrt{2}}(|\uparrow \rangle _{x}+|\downarrow \rangle _{x}) \end{aligned}$$

(4)

$$\begin{aligned} |\downarrow \rangle _{z}=\frac{1}{\sqrt{2}}(|\uparrow \rangle _{x}- |\downarrow \rangle _{x}) \end{aligned}$$

(5)

Give (ECA), if a situation that is a candidate for representing actuality makes true a proposition that captures the fact that the system is in state spin up (down) along the z-axis, then also the situation that verifies a proposition describing the system in a superposed state of spin up plus (minus) spin down the x-axis must be a candidate for representing actuality.

To sum up, when a system in the actual world is in state spin up along the x-axis, and thus in a superposition of spin along the z-axis, there are three situations that are candidates for representing actuality that Darby and Pickup explicitly accept:

s$_{1}$::

{$p_{z^{+}}$}

s$_{2}$::

{$p_{z^{-}}$}

s$_{3}$::

{$p_{z^{+}}$, $p_{z^{-}}$}

where ‘s$_n$: {$p_1$ ...$p_n$}’ is a shorthand for ‘situation s$_n$ verifies propositions $p_1$ ...$p_n$.’

Nevertheless, there are other situations that are candidates for representing actuality. The first one expresses the fact that the system is in state (2), namely, situation s$_{4}$ which makes true only ($p_{x^{+}}$). Given (ECA), if a situation that verifies only a proposition that describes the state of the system as having spin up (down) along the z-axis is a candidate for representing actuality, it follows that a situation that makes true only a description of the system as being in a superposition of spin along the x-axis also must be a candidate for representing actuality. Let us capture this fact with the following propositions:

($sp^{+}_{x}$):

The system is in a superposed state of being spin up plus spin down along the x-axis.

($sp^{-}_{x}$):

The system is in a superposed state of being spin up minus spin down along the x-axis.

According to (ECA) then, there must be two situations, s$_{5}$ and s$_{6}$, that verify, respectively, ($sp^{+}_{x}$) and ($sp^{-}_{x}$).³⁰

Therefore, the list of the possible situations that are candidates for representing actuality is the following:

s$_{1}$::

{$p_{z^{+}}$}

s$_{2}$::

{$p_{z^{-}}$}

s$_{3}$::

{$p_{z^{+}}$, $p_{z^{-}}$}

s$_{4}$::

{$p_{x^{+}}$}

s$_{5}$::

{$sp^{+}_{x}$}

s$_{6}$::

{$sp^{-}_{x}$}

Now my argument against Darby and Pickup can be easily stated. Recall again ($p_{x^{+}}$): the system is in a state that is an eigenvector of the operator $\sigma _{x}$ with a corresponding eigenvalue +1. Such a proposition seems to be determinately true, insofar as one of the premises of the whole argument is that, in the actual concrete world, our electron is in a state (1). According to the Darby and Pickup model, a proposition is determinately true if it is true in at least one situation (candidate for representing actuality) but false in no other situation. However, when we evaluate ($p_{x^{+}}$) according to Darby and Pickup’s account, it turns out to be metaphysically indeterminate. Indeed, it is true at s$_{4}$, but falsified by s$_{5}$ and s$_{6}$. When a system is in a superposed state, then necessarily its state is not an eigenstate of the corresponding operator, whose eigenvectors (belonging to different eigenvalues, as in the present context) are the terms of the superposition. Given that it is not metaphysically indeterminate in the actual world if the system is in state spin up along the x-axis, Darby and Pickup’s approach wrongly accounts this case as metaphysically indeterminate. It follows that Darby and Pickup’s account is not reliable for modelling quantum indeterminacy.

5 Is the argument conclusive?

The argument proposed hinges on some assumptions that might be resisted. In what follows, I will briefly discuss some ways of doing so. The conclusion of this section is that, in the best-case scenario, blocking my argument commits one to claims that are unlikely to be willingly endorsed by friends of MI.

Excluding the assumptions of Darby and Pickup’s view, my argument hinges on three main points; it goes without saying that the most straightforward way of resisting it is that of challenging them. These three additions to Darby and Pickup’s account are: (i) the admissibility of s$_{4}$ as a candidate for representing actuality, (ii) the claim that ($p_{x^{+}}$) is false in s$_{5}$ and s$_{6}$, and (iii) the admissibility of s$_{5}$ and s$_{6}$ as candidates for representing actuality.

(i) The admissibility of s$_{4}$:

A first way of blocking the argument above is to deny that the situation in which the system has a definite spin up along the x-axis, i.e. s$_{4}$, might be a candidate for representing actuality. The reasoning would be something along the following lines: when there is MI, one has to consider only those situations that deal with MI. Therefore, since the spin along the x-axis is determined, the only relevant situations are those that deal with the z spin. Such an argument would be not only highly counter-intuitive but it would betray the whole spirit of situation semantics. Indeed, Darby and Pickup’s account is supposed to differentiate metaphysically determined and indeterminate states of affairs. Therefore, if in the actual world it is determinately the case that p obtains, then there must be a situation at which p is true. From this, it follows that, insofar as in our example the system determinately has spin up along the x-axis, the proposition ($p_{x^{+}}$) must be true at least at a situation that is a candidate for representing actuality. Therefore, since s$_{4}$ verifies only ($p_{x^{+}}$) and does not misrepresent reality in any other regards, it must be a candidate for representing actuality.³¹ If it were not, indeed, the situation semantics proposed by Darby and Pickup would be unable to account for the fact that, in the actual world, it is determinately the case that the electron has spin up along the x-axis.

(ii) The truth value of ($p_{x^{+}}$) in s$_{5}$ and s$_{6}$:

An alternative way to resist my argument would be to insist that ($p_{x^{+}}$) is neither true nor false at s$_{5}$ and s$_{6}$. Indeed, one might want to argue that, when the system is in a superposed state, it is neither true nor false that it has the property of having spin up (or down) along the relevant axis. Insofar as ($p_{x^{+}}$) is supposed to capture such a property, then it should be neither true nor false at s$_{5}$ and s$_{6}$. Indeed, one might claim, such a reply is really natural because it makes ($p_{x^{+}}$) determinately true,³² as it should. Now, such a reply hinges on an ambiguity between the mathematical description of the state of a system and the property that the physical system instantiates. I address such an ambiguity in the final part of the next paragraph; the considerations that I make there elucidate why I take ($p_{x^{+}}$) to be falsified by s$_{5}$ and s$_{6}$. Before turning to that, I think it is worth it to point out that, even granting that ($p_{x^{+}}$) is neither true nor false at s$_{5}$ and s$_{6}$, my argument still points out two main worries about the Darby and Pickup account.

Firstly, it seems that if ($p_{x^{+}}$) is neither true nor false at s$_{5}$ and s$_{6}$, then these situations include a metaphysically indeterminate state of affairs, i.e. the electron being in a superposition of being spin up and down. How can we understand such indeterminacy? Following Darby and Pickup’s account, these situations should be made each of two other situations, one at which the system has spin up along the x-axis and one at which it has spin down. Such an understanding is suspicious for many reasons. On the one hand, the situation at which the system has spin down along the x-axis does not seem admissible, insofar as it determinately misrepresents the actual world. On the other, such a move will open an infinite regression of precisifications, which doubtfully helps us understand quantum indeterminacy.³³

Another difficulty is the following. Even granting that ($p_{x^{+}}$) is neither true nor false at s$_{5}$ and s$_{6}$, it still remains that, in their account, a determinate state of affairs in the actual world is understood as worldly indecision, so to speak, between different possible indeterminate states of affairs. The upshot of my argument is that they are committed to the view that when a system has determinately spin up along the x-axis, the world is unsettled between a superposition of spin up plus spin down along the x-axis (the situation s$_{5}$) and a superposition of spin up minus spin down along the x-axis (the situation s$_{6}$). When the system is superposed, it might make intuitive sense to claim that the world is unsettled between two definite possibilities. However, claiming that a determinate state can be understood as worldly indecision between indeterminate states seems plainly odd and obscure. Even if it is not a knockdown argument, I claim that a view with such a consequence is highly unappealing. In order to avoid these worries, rather than accepting that ($p_{x^{+}}$) is neither true nor false at s$_{5}$ and s$_{6}$ it is more likely that friends of Darby and Pickup’s account would be tempted to deny s$_{5}$ and s$_{6}$ as admissible candidates for representing actuality. We finally turn to this kind of reply.

(iii) The admissibility of s$_{5}$ and s$_{6}$:

There are three main ways of challenging the admissibility of s$_{5}$ and s$_{6}$ as candidates for representing actuality.

The first way consists of claiming that s$_{5}$ and s$_{6}$ are not possible because they determinately misrepresent the actual world: since the system is in a determinate state of spin along the x-axis, being in a superposed state determinately misrepresents the actual world. Note that if this were the case, then the converse should also hold: if a system is superposed in reality, then a determinate state misrepresents it. Consequently, Darby and Pickup’s account would instantly fail, insofar as accounting superposition as worldly indecision between spin up and spin down is the core of their explanation. In other words, if a superposed state along the x-axis misrepresents the actual world, given its mathematical equivalence with being spin up (down) along the z-axis, then there would no longer be any reason to claim that s$_{2}$ and s$_{3}$ are candidates for representing actuality. Such a reply would make my argument invalid, but at the prize of rejecting the Darby and Pickup account. Moreover, this move would be in opposition to the very way in which meta-level approaches see quantum indeterminacy. Indeed, according to these accounts of MI, when a system is superposed between spin up and down along an axis, the spin of the system is metaphysically indeterminate. Such indeterminacy is understood as it is not determinately the case that the system has spin up and it is not determinately the case that the system has not spin up (Darby 2010).³⁴ If this is the way in which defenders of meta-level accounts understand quantum indeterminacy, claiming that the proposition ($p_{\alpha }$) ‘it is determinately the case that the system is in state spin up along the $\alpha$ axis’ misrepresents a superposed state begs the question against their approach to MI. According to them indeed, when a system is in a superposed state, its logical description is ‘$\lnot \varDelta p_{\alpha } \wedge \lnot \varDelta \lnot p_{\alpha }$, where ‘$\alpha$’ is any axis and ‘$\varDelta$’ is the determinacy operator that must be read as ‘it is determinate that’.³⁵ A proposition that captures such a logical structure is not determinately misrepresented, nor determinately misrepresents, states of affairs captured by ‘$\varDelta p_{\alpha }$’ and ‘$\varDelta \lnot p_{\alpha }$’.

A second way to deny the admissibility of s$_{5}$ and s$_{6}$ as candidates for representing actuality is that of directly rejecting the validity of (ECA). I think that such a move would make Darby and Pickup’s account highly unappealing; indeed, what can possibly be the justification for rejecting (ECA)? I see why, in epistemic evaluations, for instance, the sobstitutio salva veritate might fail. For example, if Bob does not know that Alice’s dog is named ‘Nala’, then the proposition ‘Bob knows that Alice is playing with her dog’ might be true at the same time that ‘Bob knows that Alice is playing with Nala’ being false. Nevertheless, in the case at hand, we are dealing with mathematically equivalent descriptions of the very same state, and we are concerned with the metaphysical descriptions. In other words, we are dealing with a de re reading of the propositions: we are talking about the referents of the terms involved, rather than the terms themselves. And in a de re reading, ‘Bob knows that Alice is playing with her dog’ and ‘Bob knows that Alice is playing with Nala’ are also equally true. Therefore, I do not see how one can argue that a situation that verifies ‘the system is in state spin up along the z-axis’ might be a candidate for representing actuality, and one that makes true ‘the system is in a superposition of spin up and down along the x-axis’ could not: spin up along the z-axis describes the same state of ‘being in a superposition along the x-axis’. Moreover, denying the equivalence leaves space for rejecting the very first assumption of my argument, i.e. that a system with a definite spin along the x-axis is in a superposed state along the z-axis. Since such a fact is a well-established experimental result, I take every view that allows its rejection as at least suspicious.

I turn now to the last way of resisting my argument, which shows that I may have left room for ambiguities in the formulation of my argument. Such a consideration helps to develop a stronger version of the argument, in that I think it highlights what is—generally speaking—suspicious about using meta-level accounts of MI to explain quantum indeterminacy. One might want to point out that the situations used by Darby and Pickup are incomplete but definite, that is to say, they do not contain in themselves any indeterminacy of the sort. Such a claim might be used, once again, for undermining my claim that s$_{5}$ and s$_{6}$ are admissible situations: insofar as these propositions describe indeterminate states of affairs, they cannot be verified by a situation that is a candidate for representing actuality. Moreover, one might object that, insofar as Darby and Pickup explicitly claim that their situations are incomplete but definite, I have misunderstood their account all along. Such a reply, however, hinges on the ambiguity mentioned earlier, to which we now turn. The ambiguity lies between propositions that refer to the mathematical description of the quantum state and propositions that describe the properties instantiated by the quantum objects. Consider the following propositions:

($p_{x^{+}}$):

The system is in a state that is an eigenvector of the operator $\sigma _{x}$ with a corresponding eigenvalue +1.

(su):

The electron has the property of having spin up along the x-axis.

Propositions ($p_{x^{+}}$) and (su) should be, given the eigenvalue-eigenstate link, equivalent; indeed, (EEL) implies that they should always have the same truth value. The very core of (EEL) is exactly that of connecting the mathematical description of a quantum system to its metaphysical one, i.e. which properties it instantiates. Nonetheless, according to meta-level approaches, these descriptions are not equivalent. Reasonably indeed, when the latter is metaphysically indeterminate (for example when the system is in a superposition of spin along the x-axis), the former is determinately false: when a system is in a superposed state of spin, even conceding that the proposition that captures the fact that the system has spin up (or spin down) is indeterminate, the proposition that says that the state of the system is mathematically describable as an eigenvector of the corresponding operator is determinately false. In fact, propositions that refer to the mathematical description of the state of the system are never indeterminate: when the electron is in a state of superposition of its spin along the x-axis, the proposition ‘the electron is in state $|\psi \rangle =\frac{1}{\sqrt{2}}(|\uparrow \rangle _{x} \pm |\downarrow \rangle _{x})$’ is always determinately true. What is indeterminate is whether, when the system is in such a state, it instantiates the property of having spin up (down). I think that meta-level approaches are committed to accepting that propositions that describe the mathematical state of the system might be indeterminate; as I have done throughout the whole paper, I stick only to Darby and Pickup’s account in what follows.³⁶

So, I would concede that accepting that s$_{5}$ and s$_{6}$ as admissible might be suspected of begging the question against Darby and Pickup’s view: the core of their account is exactly that such an indeterminate instantiation of properties must be captured by a disagreement of possible situations; and therefore, a situation cannot contain indeterminate states of affairs. Nevertheless, this is of no help for the friends of their account because they must concede that the following proposition:

($i_{z}$):

The electron is in a state that is not an eigenvector of the operator $\sigma _{z}$; rather, the system is in a state that might be expressed as a linear combination of the eigenvectors of $\sigma _{z}$ with respective eigenvalues of $+$ 1 and − 1,

which is determinately true or false, might be verified by a situation that is a candidate for actuality. If this is so, the argument becomes even stronger than before: in the actual world the system is in a definite state of spin along the x-axis, and so it is in a superposition of spin along the z-axis. That is to say, ($i_{z}$) should be determinately true. Recall that in this case, Darby and Pickup say that there are two possible situations that are candidates for representing actuality, one that makes true that the system has spin up along z (s$_{7}$), and another that verifies that the system has spin down (s$_{8}$).³⁷ If s$_{7}$ and s$_{8}$ are candidates for representing actuality, then, given (EEL) and (ECA), I should be free to admit as candidates for representing actuality those situations that make true the mathematical descriptions of having spin up and down along the z-axis, that is to say, s$_{1}$ and s$_{2}$. However, this implies a contradiction again. In the actual world the system is in a superposition of spin along the z-axis; therefore, ($i_{z}$) should be determinately true. Given the truth of ($i_{z}$), it should follow that both ($p_{z^{+}}$) and ($p_{z^{-}}$) are determinately false.³⁸ These propositions should be determinately false because in the actual world the system is not in a state that is an eigenstate of the operator $\sigma _{z}$. Therefore, they should not be true at any possible situation (but false in at least one). According to the Darby and Pickup account though, they turn out to be indeterminate, insofar as ($p_{z^{+}}$) is made true by s$_{1}$ and falsified by s$_{2}$, where ($p_{z^{-}}$) is false at the former situation and true at the latter. By the same token, ($i_{z}$) also receives a wrong truth-value. Whereas in the actual world it should be determinately true, it turns out to be false, insofar as it is false at both s$_{1}$ and s$_{2}$, and it is not verified by any situation that is a candidate for representing actuality.³⁹

The only way, that I see at least, for resisting this last conclusion is to accept that propositions like ($p_{z^{+}}$), ($p_{z^{-}}$) and ($i_{z}$) might have an indeterminate truth value. But this amounts to accepting that the mathematical description of the quantum state (i.e. the mathematical description of the properties instantiated by quantum objects) is unreliable for drawing metaphysical conclusions. Indeed, such a claim would amount to either (a) the claim that, when a system is in a superposed state, then it is indeterminate which eigenvector represents the state (and so ($p_{z^{+}}$) and ($p_{z^{-}}$) have an indeterminate truth value),⁴⁰ or (b) a rejection of the eigenvalue-eigenstate link. Alas, these claims are worrisome for the friends of MI. (a) is dangerously close to an epistemic understanding of the formalism of quantum mechanics. As a matter of fact, the claim that mathematical descriptions do not tell us the exact properties of quantum systems has been considered the hallmark of epistemic interpretations of the quantum states. I would be happy to concede that such a view is the right one if forced by conclusions drawn by the formalism of quantum mechanics, established experiments or mathematical theorems. Nonetheless, if a view of MI would force me to accept such a position on the quantum states, I would rather move to another account of MI. And the reason for that is soon explained: examples of MI arising in quantum mechanics are drawn precisely by taking the formalism at face value. A metaphysical account that accepts that the quantum states do not describe the properties of a quantum system correctly undermines, from the beginning, the very reason for looking for quantum MI.⁴¹ The same happens with (b) since, as we have seen, (EEL) is the principle that justifies the claim that there is quantum MI in the first place. An account that accepts either (a) or (b) (or both), rejects beforehand the very idea that quantum mechanics presents metaphysically indeterminate states of affairs. As such, it is not something that an account of MI should embrace.

6 Conclusion

Under some controversial assumptions, it is possible to argue that non-relativistic quantum mechanics offers a genuine example of metaphysical indeterminacy. As we have seen, given that there are two main approaches to MI in the literature, many philosophers try to test their compatibility with quantum cases. Since the fall of the champion of meta-level approaches proposed by Barnes and Williams, many have tried to develop alternative supervaluationist theories able to cash out quantum indeterminacy.

In the paper, I have argued that the last of these proposals, namely Darby and Pickup (Darby and Pickup 2019)’s view, fails in the context of quantum mechanics. In this theory, the mathematical description of a state of definite spin along an axis is mathematically equivalent to the description of being in a superposition on another axis. I showed that, when one considers some propositions concerning the mathematical description of a quantum system, Darby and Pickup’s account assigns them a wrong truth-value. Moreover, I argued that the most straightforward ways of resisting my argument imply a rejection of one of the assumption needed to claim that there is quantum indeterminacy in the first place. As with Barnes and Williams’ view, showing that there is at least one quantum case in which their account fails is more than enough to inspire doubt concerning its tenability in the quantum arena.

I have taken Darby and Pickup’s account as an example. But such an example is interesting because it hints at the fact that there might be a reason why their account, as well as that proposed by Barnes and Williams previously, fails. And such a reason I think lies at the heart of meta-level approaches and the irremediable incompatibility of their core tenet with quantum mechanics. The interesting question then would be whether it is possible to generalize my argument to any other meta-level account of MI. If such a generalization is possible, this would mean that object-level accounts should be the preferred tool for understanding MI (which in turns would imply the question ‘Which object-level account, if any, deals with quantum indeterminacy?’). My intuition is that such a generalization of my argument is possible.⁴² The (EEL) tightly links the mathematical and the metaphysical description. Since meta-level approaches treat MI as an indeterminacy of the truth value of the propositions describing it, and (EEL) bridges these propositions in natural language to the mathematical ones, friends of meta-level accounts seem to be committed to the idea that also the mathematical description of quantum states might be indeterminate. Accepting that the mathematical description of quantum systems is not precise (i.e. we cannot take it at face value) seems dangerously close to forms of $\psi$-epistemic understanding of quantum formalism. As a matter of fact, a $\psi$-epistemic reading of the quantum state is clearly incompatible with quantum MI. Indeed, the very first reason for looking at quantum MI was that of taking quantum formalism at face value.

I think that trying to generalize the argument here proposed is essential. Given the popularity of meta-level approaches, one way of stopping the industry of introducing ad hoc modifications to Barnes and Williams’ account is that of showing once and for all that as long as one is willing to take quantum states as real, he/she must admit that quantum indeterminacy is not worldly indecision.—