Abstract

It has been widely noted that Humean supervenience, according to which everything supervenes on intrinsic properties of point-sized things and the spatiotemporal relations between them, is at odds with the nonlocal character of quantum mechanics, according to which not everything supervenes on intrinsic properties of point-sized things and the spatiotemporal relations between them. In particular, a standard view is that the parts of a composite quantum system instantiate further relations which are not accounted for in Lewis's Humean mosaic. But that suggests a simple solution: Why couldn't Lewis simply add these new relations to the supervenience basis? The aim of this article is to use Humean supervenience as a foil to spell out a feature of entanglement of general metaphysical interest: The way in which the metaphysical lessons drawn for two-party systems ramify when systems of many parties are considered. The main conclusion is that the proposed simple fix in fact results in a supervenience thesis different in kind from Lewis's, by making the relations in the supervenience basis global in a certain sense.

  • 1Introduction

  • 2Basic Argument against Humean Supervenience

  • 3A Natural Line of Response

  • 4A Different Kind of Supervenience Thesis

1 Introduction

David Lewis famously defended a position that he called Humean supervenience. In the ‘official’ formulation,

Humean supervenience is named in honor of the greater denier of necessary connections. It is the doctrine that all there is to the world is a vast mosaic of local matters of particular fact, just one little thing and then another. […] We have geometry: a system of external relations of spatiotemporal distance between points. Maybe points of spacetime itself, maybe point-sized bits of matter or aether or fields, maybe both. And at those points we have local qualities: perfectly natural intrinsic properties which need nothing bigger than a point at which to be instantiated. For short: we have an arrangement of qualities. And that is all. There is no difference without difference in the arrangement of qualities. All else supervenes on that. (Lewis [1986b], pp. ix–x)

It has often been noted that this supervenience thesis appears to come into conflict, with the way in which quantum mechanics treats ‘entangled’ systems, whose parts jointly instantiate relations that do not supervene on any intrinsic properties of the parts themselves; nor could the relations supervene on those intrinsic properties plus spatiotemporal relations. Lewis acknowledged the conflict, though was not particularly troubled, expressing confidence that his defence of Humean supervenience could ‘doubtless be adapted to whatever better supervenience thesis may emerge from better physics’ (Lewis [1994], p. 474).

Recently, however, philosophers of science have made more of the objection. Continuing to defend the position when his avowed deference to the authority of science should see him ‘abandon Separability (and hence his version of Humean Supervenience) forthwith’ (Maudlin [2007], p. 62),1 Lewis is taken to task as ringleader of the metaphysicians who ignore relevant science in their pursuit of the ‘philosophy of A-Level chemistry’ (Ladyman and Ross [2007], passim, especially pp. 24, 148–50).

Of particular concern for Ladyman and Ross ([2007], p. 22) is the metaphysicians' faith that although their assumptions about fundamental ontology may be wrong, they won't be that wrong, and won't be wrong in the kind of way that really matters. If you want to defend from the armchair the claim that there aren't really any tables or chairs, but only those little things whizzing around that make them up, what does it matter what the little things are actually like? Well, it matters because there may not be any ‘little things’ at all—it is a live possibility that the whole miniature-solar-system picture (the ‘A-level chemistry’ idea) is false. And if there are no little things then the whole metaphysical discussion is pointless (as shedding any light on what there is or how things are, at least). Now it might seem that this kind of misplaced faith is exemplified by Lewis's above confidence in the adaptability of his defence of Humean supervenience. But, on the other hand, one might think that pessimism is premature without spelling out the details of the conflict with quantum mechanics: If Lewis is to be used as an example of the perils of ignoring current science then more than prima facie incompatibility is needed; the prospects for adaptation would have to be hopeless or, at least, not trivial. And spelling out the features of entanglement that make this so is interesting in its own right. What I would like to do here, then, is to explore a bit further the details of the conflict, and highlight one feature that makes adapting Humean supervenience more than a simple fix.

The key idea behind the conflict is that quantum mechanics is in a certain sense holistic, and is therefore opposed to the reductionist character of Humean supervenience. The quantum state of a composite system is not determined by the states of its parts (if indeed the parts can be thought of as having states at all). A further, slightly more metaphysically—loaded view talks not in terms of the quantum state but of properties and relations. On this picture, the conflict with Humean supervenience becomes more vivid: relations between the parts of a composite system can be identified that do not supervene on the properties and relations in Lewis's supervenience basis, in which the only relations are spatiotemporal. Humean supervenience is opposed, in other words, to ‘relational holism’, the understanding of the metaphysics of quantum mechanics advocated by Teller ([1986]).2

Besides making the conflict vivid, however, this picture also seems to suggest an obvious reply: couldn't Lewis just add the troublesome relations to his supervenience basis? Say that what is crucial to Humean supervenience is the restriction to matters of particular fact; so long as that restriction is respected, then, the core idea of Lewis's doctrine is preserved. But surely instantiations of these new quantum relations are just matters of particular fact, albeit ones that don't supervene on Lewis's official basis. So isn't the obvious way to deal with the objection simply to drop ‘local’ from the ‘vast mosaic of local matters of particular fact’? Where the supervenience basis consisted of properties of point-like things and spatiotemporal relations, now it is going to have to include these strange quantum relations. So what?

That's not a rhetorical question. On the one hand, it is not at all obvious that this move is not allowed. The quote above certainly suggests that Lewis thought something like it would be compatible with the guiding spirit of his position. Moreover, there is no reason, a priori, to think that the confidence in the adaptability of Humean supervenience is misplaced. On the other hand, we won't know either way without the details.

So, Section 2 will briefly review a standard account of the conflict. Section 3 sets out reasons for thinking that the tweak is at least compatible with the spirit of Humean supervenience, by way of motivation for the main point in Section 4. This is that, although the resulting picture is in some respects like Humean supervenience as defended by Lewis, there is an important respect in which entanglement changes the kind of supervenience thesis involved. The same problem that seems to refute Humean supervenience in the usual examples of two-party systems ramifies for many-party systems, in such a way that it is no longer the case that the world is constructed out of binary relations between distinct objects or ternary relations, etc. The relations in the supervenience basis are themselves global in a metaphysically interesting sense.

2 Basic Argument against Humean Supervenience

The basic case against Humean supervenience, from the non-factorizability of entangled states, starts like this:3 The state of a quantum system can be represented by a unit vector |α〉 in a vector space.4 States of composite systems are represented by sums of tensor-products of the vectors used to represent states of their components. A state represented by a trivial sum |α〉 ⊗ |β〉 is factorizable. It can be thought of as factorized into a part telling us that the party whose state is represented on the left of the tensor product is in state α and a part telling us that the party whose state is represented on the right of the tensor product is in state β. A non-factorizable entangled state, like the famous singlet state forumla, cannot be written in that form. In this state, it is just not possible to assign a state vector to either party separately, on its own.

This is not yet a failure of supervenience (‘No difference in the As without a difference in the Bs’), but to get that we just have to note that actually there is a way of assigning states to the component parts. The state vector of the whole system uniquely determines a density operator which delivers probabilities, expectation values and so on in much the same way that the vector does. With |α〉 now replaced by the density operator ρ, the formula 〈α |A |α〉 for the expectation value, for observable A on a system in state |α〉 is replaced by TrA), the trace of the operator that is the product of the two operators ρ and P.

The density operator corresponding to the vector  

formula
is  
formula
To get a state for one of the parties in the two-party system considered alone, the other party is ‘traced out’. For example, the second party can be traced out of ρϕ to give forumla as follows. (The ‘1’ subscript signifies that this is the reduced state that belongs to party 1, the one whose state is on the left of the tensor product.) Wherever the second party shows up in ρϕ as something of the form ⊗ |u〉〈v|, it is replaced by the number Tr|u〉〈v|:  
formula
Since Tr|1〉〈0| = 〈0|1〉 = 0 = Tr|0〉〈1| and Tr|1〉〈1| = 〈1|1〉 = 1 = Tr|0〉〈0|, the first and second terms have a factor of one, and the third and fourth terms have a factor of zero. So:  
formula
The utility of this, the reason it can be thought of as a way of assigning a state at all to a member of an entangled pair, is that it delivers the right measurement predictions in the following sense: the expectation value Tr1A) for the result of a measurement of A on the first party, when its reduced state is ρ1, is the same as the expectation value forumla for measurement of forumla on the whole system (‘Measure A on party 1 and measure nothing on the other party’) when the state of the whole system is ρ.

Where is all this going? Well, we have succeeded in assigning some kind of state to an individual member of an entangled pair. But now consider a different composite state:  

formula
The density operator for this state is:  
formula
The crucial point is that the difference between ρϕ and ρψ is going to disappear when one of the parties is traced out. Tracing out the second party gives:  
formula
Again, since Tr|1〉〈0| and Tr|0〉〈1| are zero, and Tr|0〉〈0| and Tr|1〉〈1| are one, this equates to:  
formula
The terms that remain are the same for the two states, so the reduced density operators forumla and forumla are the same for the two different joint states. Similarly, forumla. This means that, although there is this way of assigning states to the parties on their own, there can be a difference in the joint state without a difference in the reduced states assigned to the parties separately, which is exactly what supervenience denies.

This argument can be extended to include spatiotemporal relations (so that everything in Lewis's Humean basis is the same in the reduced states whilst the joint states differ). This already strongly suggests the failure of Humean supervenience, so long as the quantum state given by the density operator is taken to be a real characteristic of the system.5 To turn these facts about the mathematical structure of the theory into a proper refutation of Humean supervenience requires more interpretation and metaphysics. In particular, we might want more talk of properties and relations in the picture, since that is the language in which Lewis's thesis is formulated. In that case, probably the most straightforward way of fleshing things out is to invoke a form of the the eigenstate–eigenvalue link: The properties that the system might possess are represented by projection operators onto subspaces of the vector space, and a system possesses a certain property if and only if it is in an eigenstate of the corresponding projector. We might be interested in two such projectors: P0 projecting onto the ray in which |0〉 lies and representing the property of being spin-up in the z-direction; and P1 projecting onto the ray in which |1〉 lies and representing the property of being spin-down in the z-direction. The projector forumla onto the whole space can also be thought of as representing a property, but a trivial one possessed by every system.

Properties of composite systems are represented by sums of tensor-products of the projectors used to represent properties of their components. In the case of those like P0 ⊗ P1, a simple product of a projector representing some property for the first party and a projector representing some property for the second party, an eigenstate with eigenvalue 1 is one in which each party instantiates the relevant property: the first party is z-spin-up, the second z-spin-down. Eigenstates of this kind of projector are important because they are the cases where we might say that the individual members of a composite system instantiate particular properties separately, all on their own, without reference to the other party in the joint state. Although P0 ⊗ P1 strictly represents a property of the whole system, that is just the property instantiated by something whose parts themselves separately instantiate the properties P0 and P1 respectively. A special case are the projectors of the form forumla. Eigenstates of this kind of projector are those where the first member instantiates the property represented by P and the second member instantiates the trivial property. Many important properties, however, do not correspond to projectors of the form P ⊗ P ′. Examples are the projectors Plike = (P0 ⊗ P0 + P1 ⊗ P1) and Popposite = (P0 ⊗ P1 + P1 ⊗ P0), which simply represent properties of the composite system or, what comes to the same thing, relations between its parts.

The non-supervenience argument then goes like this: |ψ〉 is, and |ϕ〉 is not, an eigenvalue-1-eigenstate of Popposite, while |ϕ〉 is, and |ψ〉 is not, an eigenvalue-1-eigenstate of Plike. So two parties in the state |ϕ〉 instantiate the relation represented by Plike but not the relation represented by Popposite, whereas two parties in the state |ψ〉 instantiate the relation represented by Popposite but not the relation represented by Plike. But there is nothing in Lewis's basis that the instantiation of such relations could supervene on: neither |ϕ 〉 nor |ψ〉 is an eigenstate of anything of the form forumla or forumla, and the ϕ-pair may be a duplicate of the ψ-pair as far as all of the state-independent properties (mass, charge, etc.) are concerned. So there can be a difference in instantiation of these relations without any difference in instantiation of properties by the parties considered individually (and without any difference in spatiotemporal relations—indeed, theoretically, the parties may be entangled for spin while each in a position eigenstate), which is a failure of Humean supervenience. (We know that this must be the case because both |ϕ〉 and |ψ〉 give the same reduced state when one of the parties is traced out, which means that they must be the same with respect to anything of the form forumla or forumla—the raison d'être of the reduced state is that it captures this.)

This analysis, with explicit talk of properties and relations, is by no means the only way of thinking about entanglement, but it is fairly common—relations are central in, for example, (Teller [1986]; French [1989] and Esfeld [2004]). There are differences among these, of course, in terms of how things are cashed out and the argument for failure of supervenience (emphasizing Bell's theorem versus the formalism, for example). Perhaps most significant for our purposes, though, is the fact (see Butterfield [1993], pp. 462–3) that these new relations are external in Lewis's sense of failing to supervene on the natures of the relata considered individually, but supervening on the nature of the composite of the relata (Lewis [1986a], p. 76). For Lewis, the paradigm external relation is one of spatiotemporal distance, so his formulation of Humean supervenience has just this one type of external relation in the supervenience basis (falsely, as it has now turned out).

Beyond this common way of understanding entanglement, it is tempting to ask what property is represented by the projector Plike. Well, if a system is in any eigenstate of Plike, be it |0〉 ⊗ |0〉, |1〉 ⊗ |1〉, or forumla, and z-spin is measured on both parties, then either both will be found to be spin-up (+1), or both will be found to be spin-down (−1). Whatever happens, the product of the spins will always be +1. Thus, the property represented might be that instantiated by a system in which the product of the spins is +1. Sliding between talk of properties of the whole and relations between the parts, the relation represented would be the relation of being correlated, or perhaps alike in z-spin.

In one way this is highly tendentious: to say that Popposite represents the opposite-z-spin relation implies that a system in an eigenstate of Popposite is one in which one party is spin-up and the other party is spin-down. And, assuming the eigenstate–eigenvalue link, that implies that one party is in an eigenstate of P0 and the other is in an eigenstate of P1, and that the system is therefore in an eigenstate of either P0 ⊗ P1 or P1 ⊗ P0. While that is true for some eigenstates of Popposite (e.g. |0〉 ⊗ |1〉 and |1〉 ⊗ |0〉), it is not true in general (for example, forumla). It sounds odd to say that particles are of opposite spin if neither of them even has a definite spin state.

On the other hand, one might think of it like this: In a singlet state, things are indeterminate between two states of affairs. In the first, one of the particles is spin-up and the other spin-down, and, in the second, things are reversed. Then it is indeterminate whether any given particle is spin-up or spin-down. But it is not thereby indeterminate whether the particles stand in the opposite-spin relation, because that is the case in both states of affairs. (Analogously, Teller ([1986], p. 79) is happy to talk of an eigenstate of inter-particle distance but not of position for either particle, in which case the particles would seem to instantiate the given distance relation without having determinate positions.)

In any case, this further move, to think of the relation represented by Plike as the ‘like-spin-relation’, is unnecessary to make the point. Just so long as Plike represents some relation then we have the conflict with the letter of Humean supervenience. (As noted above, even this might be unnecessary, perhaps, if things are put simply in terms of the quantum state, but the conflict is more vivid and Lewisian in the framework of properties and relations.)

3 A Natural Line of Response

That, essentially, is the argument against Humean supervenience from quantum mechanics. Maudlin, Ladyman and Ross, and others take it to more-or-less refute Lewis's metaphysics, and also to make a wider metametaphysical point—that metaphysics ought to pay closer attention to natural science.

Lewis himself considers the conflict shortly after setting out the official position. He first concedes that the thesis is only contingently true: ‘Really, what I uphold is not so much the truth of Humean supervenience as the tenability of it. If physics itself were to teach me that it is false, I wouldn't grieve’ (Lewis [1986b], p. xi). He then goes into the first stage of grief, which is denial, refusing to take lessons in ontology from a theory infected with ‘instrumentalist frivolity, […] doublethinking deviant logic; and […] supernatural tales about the power of observant minds to make things jump’ ([1986b], p. xi). And indeed the argument presented above is not quite watertight, for various reasons. Some gaps are probably just a matter of filling in the details. For example, Lewis takes the properties of point-sized things as the basis, but particles are not really point-sized, so we have not quite fully demonstrated the failure of Lewis's basis.6 Other gaps are wider, concerning substantial interpretative assumptions made in Section 2 to get the argument against Humean supervenience going, such as the assumption that every projector corresponds to a property or relation.7 In general, the result was read off from the mathematical formalism without much regard for the interpretational problems that quantum mechanics faces.

This denial does not go down well with Lewis's critics. Maudlin ([2007], p. 62) points out that there are plenty of interpretations of QM that don't deal in instrumentalist frivolity and the like, and they all appear to imply the failure of Humean supervenience in more-or-less the way presented above. Moreover, it need not be a question of taking lessons from quantum mechanics, with its interpretational problems, anyway. Bell's theorem shows that, given fairly minimal realist assumptions, nonlocality would have to infect any theory which reproduces the right statistics, statistics which have since been experimentally verified (the Aspect experiment). Along these lines, Ladyman and Ross ([2007], p. 149, fn. 36) claim that ‘Bell's theorem tells us that any […] resolution [of the measurement problem] must do violence to at least part of Lewis's metaphysical picture’.

Now, Lewis does have plenty of room for manoeuvre among the various interpretations of the formalism and the various metaphysical spins that may be put on the same interpretation. One example is the suggestion of Loewer ([1996]) that something like Humean supervenience is restored if the fundamental space of the world is configuration space (which offers, for a range of solutions to the measurement problem, a fundamental ontology on which the non-supervenience problem seems to disappear). Likewise, Bell's theorem itself is amenable to a variety of interpretations, depending on what is meant by nonlocality. To interpret it as threatening Humean supervenience involves what might be thought of as a metaphysically heavy reading (non-supervenience), rather than a kind of action-at-a-distance that would be relativistically problematic but arguably metaphysically innocuous (at least as far as Humean supervenience is concerned). Here substantive interpretive choices have to be made about, for example, decomposing Bell's locality assumptions in a way that is supposed to reflect those metaphysical theses (see Jones and Clifton [1993], for opposition to this line), and Lewis would not be alone in resisting the conclusions needed to threaten his position. So a Lewisian might stick to (something like) the letter of Humean supervenience by arguing for a particular solution to the measurement problem, along with a suitable metaphysical interpretation of that solution.

But exploiting this wiggle-room serves, as far as the critics are concerned, only to illustrate the wider methodological complaint. The configuration-space manoeuvre, for example, constitutes ‘the ultimate elevation of Separability as a regulative principle, rather than an empirical theory, and urges even more strongly the question of motivation’ (Maudlin [2007], p. 61). No doubt much more could be said in reply—for example, Loewer's suggestion follows a general argument (Albert [1989]) for the configuration-space view, which has nothing to do with defending Humean supervenience—but given Lewis's disavowal of ‘reactionary physics’ (Lewis [1994], p. 474), an alternative to this kind of rearguard action would better suit the dialectical position.

So, rather than denial, acceptance might be a better option. Presumably this is what Lewis has in mind when he says that Humean supervenience can ‘doubtless be adapted’. And, in fact, one might think that acceptance is easy: simply add the offending relations to the supervenience basis (or perhaps have them replace the previous—spatiotemporal—relations altogether). Clearly, this takes us some distance from the official statement of Humean supervenience, but how far, and in what direction?

One way to violate the letter of Humean supervenience—one direction of departure—would be the addition of lawmaking relations between universals. To add those to the supervenience basis is to add something over and above matters of particular fact, local or not. This seems to take us a considerable distance from Lewis's original position. Given the motivation (from the Introduction to Lewis [1986a]) of showing that these metaphysically puzzling things can be done without, each and any addition of this kind makes the supervenience thesis much less interesting. (In any case, Lewis thinks the additions pointless because he doesn't see how they could possibly do their jobs—you can't just call the addition ‘necessitation’; likewise, for un-Humean chancemakers (Lewis [1986b], pp. xii, xv–xvi).)

The addition of further external relations, however, in response to quantum mechanics seems like a quite different kind of violation of the letter of Humean supervenience. One kind of external relation—spatiotemporal—is already found in the supervenience basis, so if the addition amounts to just more of the same, then one might think it fairly innocuous. Lewis's aim is ‘to resist philosophical arguments that there are more things in heaven and earth than physics has dreamt of’ (Lewis [1994], p. 474), but if physics itself dreams of relations of entanglement then adding them is perfectly well-motivated. The effect of altering Humean supervenience in this direction—how far away it takes us—would depend on the nature of the relations, and this is what will be considered in Section 4.

So, the letter of Lewis's Humean supervenience can be violated along various dimensions, and to varying degrees. It also seems, though this is less obvious, that some dimensions are more significant than others. The ‘more of the same’ thought in relation to external relations mirrors Lewis's comfort with the plurality of possible worlds, which he justifies with the famous distinction between qualitative and quantitative ontological parsimony, or between cost in ontology versus ideology (Lewis [1986a], p. 4). This line of thought is what suggests a spirit of Humean supervenience. Corresponding to the ‘broadly Humean doctrine’ of Lewis ([1980, p. 111)—which ‘holds that all the facts there are about the world are particular facts, or combinations thereof’ (and so differs from Lewis ([1986b], pp. ix–x) in saying nothing further about the nature of the particular facts)—this would allow variation along some dimensions (for example, differences in the nature of spacetime and the external relations that unify it), but would be violated by moving in others (the addition of unHumean chancemakers or lawmaking relations between universals). The thought then is that the addition of relations to the supervenience basis in response to entanglement violates the letter, but not the spirit, of the doctrine.

(Other metaphysical lessons from quantum mechanics are a different matter. It might be, for example, that the best way of interpreting the theory includes dispositions or some other uncomfortable facts that don't find a place in the categorical basis (if so, then the relations considered here might also be dispositional). In that case, things would already be bad for even the spirit of Humean supervenience before multi-party entanglement is considered. But that would be a separate issue to the present one (though ultimately pulling in the same direction).)

Although the distinction between letter and spirit seems fairly clear—in particular, the broadly Humean doctrine (Lewis [1980], p. 111) looks very like the official Humean supervenience thesis (Lewis [1986b], pp. ix) minus the requirement that the matters of particular facts be local—it need not be hard and fast. For one thing, while he is explicit about the kinds of things that are to be constructed from, rather than added to, the supervenience basis, it is not as though Lewis would never countenance adding a primitive. For example, he is prepared at the right price to break with an egalitarian approach to properties in Lewis ([1983]). So it does not seem that movement from the official position would be confined once and for all to some directions and not others. For another thing, to draw a firm line around the spirit of Humean supervenience would require a secure and generally applicable distinction between alterations that merely change the matters of particular fact, and those that do something else. With the letter of Lewis's specific doctrine in mind, we have a good handle on what is and is not a local matter of particular fact. But the wider context requires a general specification of matter of particular fact, which is less straightforward.8 In a similar vein, the ‘more of the same’ thought is more problematic than it seems—see (Melia [1992]) on Lewis's use of the qualitative/quantitative distinction.

Trouble drawing a clear boundary around the spirit of Humean supervenience might just show that the spirit is vague, not that there is no such thing, and the interesting questions remain: if relations are added to the supervenience basis in response to entanglement, how far do we move from the letter of the official account, and why is the change significant? So in Section 4 I describe one way in which the addition of the extra relations crosses a line and takes us a certain conceptual distance from (Lewis [1986b], pp. ix). Wherever this line lies in relation to the boundaries of the spirit of Humean supervenience (if there is a boundary), and whether or not the line is metaphysically significant for Lewis, it is certainly metaphysically interesting.

4 A Different Kind of Supervenience Thesis

Suppose we have a division of the world into multiple parts.9 Then we can distinguish various levels of supervenience thesis: Lewis's working assumption seems to be that there is a set of point-like parts, and that the monadic properties of those, plus spatiotemporal relations, constitute an adequate supervenience basis. Spatiotemporal relations are also naturally viewed in Lewis's metaphysics as binary, so Humean supervenience comes in at Level 2. This nicely fits the mosaic metaphor: the world is cobbled together piece-by-piece by connecting parts with the spatiotemporal relations. If you want to add another piece to the mosaic, you just fix its binary relations to each of the other parts, one-by-one.

  • Level 1: Everything supervenes on monadic properties of the parts.

  • Level 2: Everything supervenes on monadic properties of the parts plus binary relations between them.

  • Level 3: Everything supervenes on monadic properties of the parts plus binary and ternary relations between them, etc.

Can things still be thought of like this once the quantum relations are added to the supervenience basis? At first glance, thinking of Plike and Popposite, one just might hope that they can. Thinking of alikeness and oppositeness in general, if A is like B, and B is like C, then it follows that A is like C and that A, B, and C must instantiate the three-place alikeness relation. Translating into the language of projectors (and noting the concerns about thinking of a ‘like-spin’ relation), if the system is in an eigenstate of forumla, and also an eigenstate of forumla, then it is in an eigenstate of forumla, and of P0 ⊗ P0 ⊗ P0 + P1 ⊗ P1 ⊗ P1. Here is an example of a ternary relation whose instantiation is fixed by fixing the binary relations, just like the instantiation of the three-place ‘standing in a one metre equilateral triangle’ relation is fixed by the pattern of instantiation of the binary ‘one metre apart’ relation. If it always works like this then the new supervenience thesis is also at Level 2, and still fits the mosaic metaphor: it is still possible (subject to the success of the philosophical arguments for Humean supervenience) that the world is cobbled together piece-by-piece by connecting parts with binary quantum–mechanical relations. Again, if you want to add another piece to the mosaic, you just fix its binary relations to each of the other parts, one-by-one.

Unfortunately, this doesn't work in general. Start with a generalization of the standard argument, where instead of two parties we have n. The state:10 

formula
gives the density operator  
formula
while the state  
formula
gives the density operator  
formula
Again, different joint states but, again, the difference lies in the third and fourth terms |1〉〈0|⊗ n and |0〉〈1|⊗ n, which can't make a difference to the reduced states because they will disappear. Even tracing out just one party from either ρ or ρ′ gives the operator  
formula
(because, again, the third term in either ρ or ρ′ is multiplied by Tr(|1〉〈0|) = 0, and similarly for the fourth). That goes whichever party is traced out: all n of the (n − 1)-party reduced states are the same for ρ and ρ′. Likewise all the (n − 2)-party reduced states, and so on. So even the totality of all the reduced states is not enough to determine the joint state of the whole n-party system.

This translates into a non-supervenience thesis about properties and relations as follows: Information about whether the entire n-party system instantiates a certain property, or whether all of its parts instantiate a certain relation, is captured by whether or not the system is in an eigenstate of a certain projector. This projector has the form of a sum of n-fold products P1 ⊗ P2 ⊗ P3 ⊗ … ⊗ Pn. We want to know whether the instantiation of those n-place relations is fixed by the instantiation of some set of relations by collections of fewer than n of the parts. If we are hoping for a Level-2 supervenience thesis, then we want to know whether the instantiation of those n-place relations is fixed by the instantiation of some set of binary relations by pairs of the parts.

Consider then some set of fewer than n parts; for concreteness take some n − 1 of them. Information about whether or not these n − 1 parts instantiate some n − 1-place relation is captured by whether or not the whole system, in state ρ, is an eigenstate of a certain projector. This projector has the form of a sum of n-fold products forumla., with the identity in the place corresponding to the part that we're not considering (the first, in this example). But whether or not that is the case is also fixed by whether or not the reduced state ρreduced for those n − 1 parts is an eigenstate of the projector that results by removing forumla from each term in the previous sum. That is, we are now interested in the sum of (n − 1)-fold products P2 ⊗ P3 ⊗ … ⊗ Pn. (As before, capturing this is the point of the reduced state.)

In general, whether or not some collection of parts instantiates some relation is fixed by the appropriate reduced state. So, fixing all the reduced states fixes all of the relations of less than n places instantiated by collections of parts. If fixing all the relations of less than n places fixed everything whatsoever, then fixing all the reduced states would fix everything whatsoever. But the reduced states don't fix everything (that was the generalisation above, where systems are in different joint states ρ and ρ′, and so instantiate different n-place relations, but the reduced states are exactly the same). So fixing all the relations of less than n places doesn't fix everything either.

It is not true, then, that all else supervenes on anything less than the intrinsic properties of the whole system. The n-place relations instantiated by the members of an n-party system are going to have to go into the supervenience basis. Since there is no limit to the number of parties that can be involved in an entangled state, there is no limit to the adicity of the relations involved. Therefore, none of those levels above is enough to characterize the kind of supervenience thesis that results.

Where does this leave Humean supervenience? Well, on the one hand, the supervenience basis still consists only of matters of particular fact. So, tentatively, I suggest that there is no conflict with the spirit of the principle. But we haven't just made a slight alteration to the letter. It's not just that relations besides spatiotemporal ones have to be included in the supervenience basis, which one might think of as a fairly mild kind of nonlocality.11 Rather, there is a sense in which this picture is as diametrically opposed to locality as it could be—the local matters of particular fact have been replaced by global matters of particular fact. It is not just a matter of what the natural external relations happen to be, but a difference in how they behave: it is no longer possible to think of the world as being built up with these relations one component at a time. This marks a clear departure from Lewis's ‘vast mosaic’ and the piecewise construction metaphor that it evokes.

Funding

Leverhulme Trust (ECF/2010/0228).

Acknowledgements

For comments and discussions, I would like to thank Laurence Goldstein, Phyllis McKay Illari, Brian Weatherson, and Jon Williamson. I am especially grateful to three anonymous referees for the BJPS, who provided very detailed and helpful comments.

1 ‘Separability’ has a number of related uses. In Maudlin's sense it is the doctrine that ‘[t]he complete physical state of the world is determined by (supervenes on) the intrinsic physical state of each spacetime point (or each pointlike object) and the spatio-temporal relations between those points’ (Maudlin [2007], p. 51).
2 Although ‘relational holism’ is specifically Teller's term, there are many similar these. For example (Healey [1991]). The intention here is not to restrict attention to a particular formulation, but to consider the general set of ideas that are increasingly being cited as problematic for Lewisan metaphysics.
3 This is how Maudlin ([1998]) challenges a particular Humean piece of Lewis's modal realism (the recombination principle), rather than the 1986a statement of Humean supervenience per se, though elsewhere (Maudlin [2007]) it is explicitly Humean supervenience that is threatened.
4 See the Appendix to (Maudlin [2002]) or Chapter 2 of (Albert [1992]) for self-contained and accessible introductions to this; see (Hughes [1989]) or (Nielsen and Chuang [2000]) for more detail.
5 Thanks to an anonymous referee here.
6 Thanks to an anonymous referee for pointing this out.
7 See, for example, (Daumer et al. [1997]) for criticism of this assumption.
8 Thanks to an anonymous referee for pressing this, and apologies for not providing the definitive answer they might have wanted.
9 Fixing the division is necessary because it's uninteresting (in a Lewisian framework anyway) that everything supervenes on the monadic properties of the mereological sum of everything.
10 |0〉n just means |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ ‥, n times
11 After all, the spatiotemporal relations already go beyond pure monadic properties instantiated by point-sized things, so even Lewis's original picture doesn't quite fit his own sense of locality (thanks to an anonymous referee for pointing this out).

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