Truthmaking and Analytic Combinatorialism
Roughly, the principle of recombination says that anything can coexist and fail to coexist with anything else. But that's too strong: things do have essential extrinsic properties; if Kripke's origin is essential to Kripke, Kripke cannot fail to coexist with his ancestors. However, a perfect intrinsic duplicate of Kripke could fail to coexist with Kripke's ancestors. So less roughly, the principle of recombination goes somehow like this:
For any things in any possible world there is a world which contains any number of perfect intrinsic duplicates of all those things and nothing else (i.e. nothing distinct from all these duplicates).
What is a perfect intrinsic duplicate? Something that has exactly the same intrinsic properties as the original. What is an intrinsic property? A property that belongs to objects independently of what exists and goes on around them. The instantiation of an intrinsic property in some region of a world is independent of the instantiation of intrinsic properties in other regions.
Suppose the recombination principle is false because there are things x, y such that no world contains perfect intrinsic duplicates of exactly one of them. Then x has an intrinsic property P and y an intrinsic property Q such that necessarily, if something has P then there is something else distinct from that thing which has Q. But this is impossible if P and Q are really intrinsic.
It is equally impossible that for some things x, y, no world contains perfect intrinsic duplicates of both. At least for finitely many duplicates of finitely many things, the recombination principle is analytic. I'm not quite sure what happens in some infinite cases. Suppose there is a limit to the size of possible worlds. Then the principle is false and there is an intrinsic property P such that necessarily, if something has P, it does not coexist with k many other things having P. Is that coherent? Maybe yes, because not coexisting with k many Ps is a necessary property. Surely having two parts is not extrinsic just because necessarily, if something has two parts it does not coexist with a round square.
Anyway, Lewis seems to have thought that the principle is much stronger than it actually is. For example, in "A world of truthmakers?", he argues that we face a choice between combinatorialism (the recombination principle) and the truthmaker principle: by the truthmaker principle, there is something x such that necessarily, x exists iff there are no arctic penguins; but by the recombination principle, anything can coexist with anything, so x can coexist with arctic penguins.
The apparent conflict already happens with finitely many things for which the principle is analytic. So if this conflict really exists, the truthmaker principle is analytically false. But plainly, there is no conflict here at all. The recombination principle does not say that anything can coexist with anything. It only says that an intrinsic duplicate of x, the truthmaker of "there are no arctic penguins", can coexist with intrinsic duplicates of arctic penguins. It doesn't follow that x itself can coexist with arctic penguins. For one, the intrinsic duplicates of the arctic penguins could be antarctic penguins. And in general, why couldn't x be essentially such that there are no arctic penguins?
The recombination principle is irrelevant here. The issue is not truthmaking versus combinatorialism but truthmaking versus very extrinsic counterpart relations. Lewis probably realized this by the time he wrote "Things qua Truthmakers". There he argues that not coexisting with arctic penguins is not a kind of similarity that could relate counterparts. This claim, not the recombination principle, is what proponents of truthmakers should reject. In my view, that is no cost at all. Lewis doesn't offer any reason for his restriction to the counterpart relation. In fact, the counterpart relations he uses for doxastic counterparts and counterparts by acquaintance seem to be just as extrinsic.