Humean Confusion
I'm confused. It seems to me that the Dense Worlds Argument refutes Lewis' Humean Supervenience thesis: Not all facts about worlds without alien properties are determined by the distribution of fundamental properties over space-time points. But that's not what really worries me.
What worries me is that I don't know what to blame. I don't see any move that doesn't lead into further difficulties. Consider blaming HS. If HS is false, then our world either contains extended things (as opposed to points) that instantiate fundamental properties, or it contains things that stand to each other in fundamental but not spatio-temporal relations. Let's focus on the second possibility. It is certainly conceivable that fundamental properties are instantiated by extended things. But does this help? Suppose all fundamental properties are instantiated by cubes with a volume of 1 nm^3 (or stages of such cubes with volume 1 ns*nm^3). Then the same kind of shuffling as in the dense worlds argument still shows that the interesting facts about our world are independent of the distribution of fundamental properties. But this time HS is not among the assumptoins, so we can't use the argument as a reductio against it.
The problem is that one of the following must go:
1) There are some properties such that all facts about our world are determined by the spatio-temporal distribution of these properties. Let's call them "fundamental properties".
2) Some alternative possible worlds (distinguishable from ours) can be reached by rearranging the stuff of our world, that is, by rearranging the distribution of fundamental properties.
3) All fundamental properties are intrinsic. So whether a thing instantiates such a property is independent of the role the thing plays in the world.
4) Hence by the principle of recombination, there are worlds indistinguishable from our world where the fundamental properties are differently distributed.
5) Some of these worlds have exactly the same distribution of fundamental properties as the worlds mentioned in (2).
It follows that the same distribution of this-wordly properties can result both in a world that is distinguishable from ours and in a world that is indistinguishable from ours. And this just makes nonsense of the assumption that the distribution of this-wordly properties determines all the facts about our world.
On first thought, the most obvious candidate to give up is (5), but I think it is strongly supported by arguments like dense worlds. To avoid (5), one would need heavy restrictions upon possible distributions of fundamental properties, that is, restrictions upon recombination. For example, if by a primitive modal law some property P can only be instantiated by something that contains a proper part instantiating Q, this would prevent certain recombinations. In particular, P and Q could not trade places. They also couldn't trade places if by a primitive modal law P can only be instantiated at triangular objects, and Q only at rectangular ones. But there would have to be an ubelievable amount of such primitive modal laws to avoid (5).
To illustrate, consider the two books on my desk, a red one lying on top of a blue one. (As it happens, they have exactly the same size and shape.) There is possible world like ours except that for a single second those two books change places. Obviously, this world can be reached by rearranging the stuff (the books, in fact) of our world. Let's call it R. R is also a world where the distribution of fundamental properties differs from the distribution over here because the rearranged books and their parts take their intrinsic properties with them.
But those intrinsic properties do not all by themselves determine how the rearranged books interact with other things (including how they look to us). So there is a further world Q where the distribution of fundamental properties is exactly the same as in R but where the fundamental properties of the rearranged books and their parts play different roles for a second -- namely the roles that in our world the corresponding properties of the other book plays. For this second, the blue book and its parts behave exactly like the red book and its parts ordinarily behave, and conversely. Q is therefore indistinguishable from our world, even though it has the same distribution of fundamental properties as R, which is distinguishable from our world.
If (5) is to blame, it has to be metaphysically impossible for the fundamental properties of the two books and their parts to trade places in the way described. But why should that be impossible?
If it was impossible then much of an intrinsic property's role would be essential to the property: The intrinsic properties of the blue book could not possibly play the roles of the corresponding properties of the red book. That is, whether or not something can have these properties depends on how other things interact with it. And this just refutes that the properties are intrinsic in the first place.
So it looks like we should rather give up (3): Fundamental properties needn't be intrinsic. There's something very intuitive about this, because fundamental physical properties don't look intrinsic at all: Having spin-up just means to behave in a certain way. Intuitively, it is quite absurd to suggest that some particle might have spin-down even though it behaves like spin-up. But I'm afraid this leads into even worse difficulties. For one, as Smith and Stoljar have argued in The Monist 1998, it may leave us postulating bare similarity relations between worlds. Which is ludicrous. Moreover, it now looks as if if the fundamental properties of a thing are determined by how it interacts with other things. Which simply means that those properties aren't fundamental.
[Update (2 hours later): Well, no, it obviously doesn't mean that. It is possible that everything is determined by the A-facts even though the A-facts themselves are determined by the B-facts. I still don't think this is a way out.]