A better Principle of Recombination?
The principle of recombination states what other possible worlds there must be, given the existence of some possible worlds. In sec. 1.8 of On the Plurality of Worlds, David Lewis suggests something like this:
L) For any parts of any worlds there is some world containing any number of duplicates of all those parts, and nothing else , provided that they all fit into a possible space-time.
Daniel Nolan argues in "Recombination Unbound" that the clause 'and nothing else' should be dropped, because if some thing B consists of two duplicates of A, there couldn't be a world containing one B, one A, and nothing else. Unfortunately, without the clause the principle doesn't exclude the necessary coexistence of distinct possibilia. In fact, it is even compatible with all possibilia having duplicates in all worlds. I think it would be better to leave the clause and instead restrict the principle to distinct parts of worlds.
Nolan also argues that the proviso about space-time should be dropped, mainly because it is unnecessary and implies a mysterious limit on possible instances of spatiotemporal relations. I'm not sure about both of these issues. As for the second, many of us used to think that there is a set of all possible worlds, and a set of all possible individuals. But if there is, then this set has a cardinality K. Hence K is a necessary limit on possible instances of any property and relation whatsoever. Anyway, Nolan suggests this:
N) For any parts of any worlds there is some world containing any number of duplicates of all those parts.
As Nolan notes, this leads to the conclusion that for any object and any K, there is a world containing 2^K duplicates of the object. Hence there is no set of all possible objects, no set of all duplicates of Mona Lisa, and probably also no set of all possible worlds. That may have advantages, in particular for nominalism about mathematics, but it destroys or complicates many of the things possibilia were supposed to do. For instance we might want to use ordinary set-theoretical functions from possible worlds to sets of things as semantic values. That's impossible if the worlds are a proper class, and some of them even contain proper-class many things.
This problem is due to the 'any number of' part in (L) and (N), and I wonder if we shouldn't just drop that. Its intuitive motivation is that if Mona Lisa exists, then there might well be two duplicates of her, or three, or infinitely many. But we get much of this from the remaining principle anyway: Since besides Mona Lisa there exists a lot of other stuff, there are a lot of different worlds containing Mona Lisa and different parts of other stuff. Actually, if space-time is real and any part of space-time qualifies as part of a world ('object' in Nolan's terms), then there are presumably at least 2^aleph_0 such worlds. So there are at least 2^aleph_0 distinct Mona Lisas (on in each of these worlds), and by the recombination principle, there is some world containing distinct duplicates of all of them. So here, finally, is my suggestion:
For any distinct parts of any worlds there is some world containing duplicates of all those parts, and nothing else.