The thesis of Humean Supervenience (HS) can be summed up by the following quotation from David Lewis:
Humean Supervenience […] 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: ix-x).
As Weatherson claims, HS can be seen as composed by two distinct thesis:
The first is the thesis that […] all the truths about a world supervene on the distribution of perfectly natural properties and relations in that world. The second is the thesis that the perfectly natural properties and relations in this world are intrinsic properties of point-sized objects, and spatiotemporal relations. (Weatherson 2016: §5).
On the one hand, the first thesis can be summed up by the slogan “truth supervenes on being”. It concerns broadly the nature of the minimal basis of entities which grounds all the truths about a world. Here, we can take ‘perfectly natural’ as synonymous to ‘fundamental’. Lewis, in fact, takes ‘perfectly natural’ properties to be those properties whose distribution entail all the truths about a world.Footnote 3 If two possible worlds contain exactly the same natural properties and relations, arranged in the same way, then those worlds will be identical – or, better, exactly similar. Thus, the first thesis composing HS amounts to the claim that there is a minimal set of (kinds of) entities which entails all the truths about a world.
On the other hand, the second thesis specifies which kinds of entities fall into the minimal supervenience basis for the actual world. There are three ingredients there: point-sized objects; those intrinsic properties instantiated by them which are also perfectly natural; and spatiotemporal relations between those point-sized objects. However, Lewis is not clear about the meaning of the term ‘point-sized’. In fact–as Weatherson (2015: 113) points out–when Lewis discusses about HS, he sometimes talks about intrinsic properties of point-sized objects and sometimes refers to local properties instead. However, these are not the same thing. On the one hand, we can read the expression ‘intrinsic properties of point-sized objects’ as referring to intrinsic properties of mereologically simple objects–as thus read ‘point-sized’ as ‘mereologically simple’. On the other hand, as Butterfield (2006) argues, we can use the term ‘local properties’ to refer to properties that supervene on intrinsic features of arbitrarily small regions. And so, we could define HS in terms of intrinsic properties of arbitrary small regions. Both interpretations seems viable, and I will not be committed to one or another in this paper. Thus, in the following, when I will talk about ‘point-size’ or ‘atomic’ objects, this terms can be taken as referring to both mereologically simple objects and arbitrarily small regions of space.
One interesting point is that, according to HS, there is only one kind of relation that is perfectly natural and therefore fundamental (and, thus, present in the supervenience basis): spatiotemporal relations. Also, only intrinsic and local kind of properties are admitted. Lewis defines ‘intrinsic’ properties–modally–as follows:
The intrinsic properties of something depend only on that thing; whereas the extrinsic properties of something may depend, wholly or partly, on something else. If something has an intrinsic property, then so does any perfect duplicate of that thing; whereas duplicates situated in different surroundings will differ in their extrinsic properties. (Lewis 1983: 197).
At this point, a natural question arises: what is the criterion we should use to choose which properties and relations can be admitted to the supervenience basis and which can not? Let us focus on relations. Keep in mind that I will use the terms ‘relation’ and ‘extrinsic property’ interchangeably, and in the following I will formally express relations in terms of extrinsic properties. However, this is just a syntactical matter, and of course no substantive argument relies on this.
Lewis claims that the final criterion for picking out which kind of relations are fundamental should come from physics. In fact, he chooses spatiotemporal relations due to their–alleged–fundamental role in physics. However–taking a step back–are there a priori criteria for choosing which relations we should take as fundamental? That is, are there criteria which have to be satisfied independently from a posteriori physical reasons?
Some philosophers have pointed out some minimal a priori criteria which every relation should meet to be considered as part of a minimal supervenience basis.Footnote 4 Let us start by considering two candidates extrinsic properties that are usually ruled out from the properties of the supevenience basis. The first property is described by Sider (2003: 140), while the second is mentioned by Merricks (1998: 60):
“Being part of a whole with such and such properties”
“Composing a square red object”
Actually, while the first is meant to be an extrinsic property, the second is expressed by Merricks as a relation that can hold between atoms. However, as I mentioned, a relation can be easily and uncontroversially translated as a (monadic) extrinsic property. Formally, this can be done via lambda abstraction, and this is precisely what I will do now. However, once again, keep in mind that this is just a syntactical move–the role of lambda abstraction is just that of making the present analysis more precise.
Thus, let us first translate 1 and 2 as follows (where F is any property whatsoever that is possessed by the whole and not by any of its parts):
“x is such that it is part of a z that is F”
“x is such that together with y composes a F whole”Footnote 5
We can then try to translate these statements in the following formalized form (where P is the parthood relation and ‘x + y’ is shorthand for ‘x and y compose’ z):
λx(∃z(Pxz ∧ Fz))
λx(∃y∃z(x + y = z ∧ Fz))
Put in this way, 1 and 2 express two properties that can be instantiated by a certain object x, which are both extrinsic properties according to Lewis’s definition above (“the extrinsic properties of something may depend, wholly or partly, on something else”).
As I have pointed out, Lewis includes only ‘perfectly natural’ or ‘fundamental’ properties within the supervenience basis linked to HS. Thus, it is quite obvious that this kind of properties would be ruled out from the basis in any case. However–and this is the crucial point–here I am claiming that, according to Merricks and Sider, we have a priori reasons for not considering these properties, even in principle. That is–independently from any consideration of naturalness–this kind of properties cannot be considered as part of the supervenience basis, because they violate an implicit criteria which any version of supervenience thesis have to satisfy–i.e. any theory which tries to posit a minimal set of truths or entities upon which every truth supervenes on.
In fact, the problem with them is that they would make HS a trivial thesis:
It is trivial to say that the existence of a square red object composed of atoms supervenes on atoms’ standing in the composing a square red object relation. (Merricks 1998: 60).
Abstracting away from the particular case, the problem with these relations is that among their relata there are not only point-sized objects (or arbitrarily small regions), but also composite wholes. Take 1 for example: λx(∃z(Pxz ∧ Fz)). This is a parthood relation between a point-sized object–which we can also call an ‘atomic’ object–and a composite whole instantiating certain properties. On the contrary, as Merricks argues, when we talk about relations within the supervenience basis, we should restrict our commitment to restricted atom-to-atom relations–i.e. relations between point-sized objects.
Why only atom-to-atom relations? Because, if we add to the basis relations which are not atom-to-atom, we would end up adding to the supervenience basis information about non-point-like objects which are supposed to supervene on the properties of spacetime points–and this is a dangerous slippery slope. In fact, if we were to add a kind of relation which is not ‘atomic’, we could in the same way add tons of them, up to the point where we would have every relation whatsoever in the supervenience basis. And, in that case, it would become trivial to say that all the truths about a world supervene on the distribution of the properties of the points of the mosaic and the relations between them, because we would already have ‘all the truths’ contained within the minimal basis.
Therefore, we can conclude that ‘being atom-to-atom relations’ is an a priori necessary condition which relations have to satisfy to be considered as parts of the supervenience basis. In the following, we shall see how–in certain situations–entanglement relations violate this necessary condition.