In Counterpart Theory and Quantified Modal Logic, Lewis sets out his counterpart theory using four primitive predicates:
Wx (x is a possible world)
Ixy (x is in a possible world y)
Ax (x is actual)
Cxy (x is a counterpart of y)
A series of postulates then guarantees, among other things, that every possible thing (everything in the range or domain of C) is in exactly one possible world, and not (in the relevant sense of in) in anything else (see Philosophical Papers, vol. 1, pp. 267).
In On the Plurality of Worlds, he goes to work eliminating all of these primitives. For our purposes the important two are W and I, which he eliminates in two steps. First, he defines both in terms of two other relations: the partwhole relation and the worldmate relation. To be in a possible world is taken to be nothing more or less than to be a part of it, in the (supposedly) ordinary mereological sense, and a world is defined as a maximal whole of parts which are worldmates. Then he tries to eliminate the worldmate relation, in turn, in favor of something like spatiotemporal relations. My question is about the motivation for that second step.
The second step lands Lewis in some difficulty, as he is well aware. The rough idea is to say that two things are worldmates if and only if they are spatiotemporally related. But this would have a consequence Lewis regards as unwelcome: it would become a necessary truth that all non-simple worlds are internally related by what we call spatiotemporal relations; that is, by the ones we actually have at this world; that is (we think) by relativistic spatiotemporal relations. Lewis wants to allow that at some worlds (e.g., a Newtonian world), the relations might be different (p. 74). So instead he proposes that any relation counts as spatiotemporal if it satisfies certain criteria, including at least: naturalness, pervasiveness, discriminativeness, and externality (pp. 756).
But this has a further consequence which strikes me as odd.
It will still be a necessary truth that something like our geometry (in a very general sense) holds at every non-simple world. That is: if a possible thing is not simple, then all of its parts must be related by something like our spatiotemporal relations.
Technically this becomes an analytic truth (it follows from the counterpart-theoretic definition of possible [thing]). But in effect it becomes synthetic a priori, insofar as it is an independent source of what Lewis calls the obscurity of our modal language (Counterpart Theory, Phil. Papers 1:29). Since the first three criteria are vague, it will be vague (therefore obscure) whether some things are possible, not because we are uncertain if we are describing them consistently, nor because we are uncertain what to count as a counterpart of what, but because we are uncertain whether to count all their parts as belonging to the same world.
Here is what Lewis could have done instead. First, define the partwhole relation in terms of the worldmate relation and the relation of exact duplication: say that x is part of y just in case every exact duplicate of y has as a worldmate some exact duplicate of x. This makes Lewiss principle of recombination (On the Plurality of Worlds, pp. 8791) come out true by definition. Then, if a broad definition of quasi-spatiotemporality seems desirable, it could go this way: spatiotemporal relations are relations that hold between things just in virtue of the fact that they are parts of the same whole. This would be similar to what Leibniz and followers mean by defining space as the order of simultaneous possibles.
Quasi-spatiotemporal relations would still have to be external (because of the just in virtue of provision), but the other criteria would be eliminated. There would be no obscure rule against any kind of geometry, no matter how exotic.
This would not get back Lewiss unrestricted mereology. It would eliminate so-called trans-world individuals, but Lewis doesnt much like those, anyway. It would also mean that wholes composed of sets (and other non-individuals) would not be wholes in the usual sense, which might be more serious for Lewis (in view of Parts of Classes), although it accords well with common sense. Still, this way of proceeding seems sufficiently attractive that Lewis ought at least to have considered it.
My rough guess as to why he doesnt is as follows. Although the principle of recombination, going this way, would be an analytic truth, there would be no guarantee that what can be recombined is (at least) spatiotemporal parts. It would be formally OK with this theory if our world were simple, for example, in which case there might be no other possible worlds (call this Spinozan Supervenience).
Against this, Lewis would need to appeal to common sense, pre-philosophical notions of compossibility. He needs to appeal to these on his way of going, as well: his non-analytic principle of recombination ultimately gets accepted on the same basis. But on the suggested alternative route, he would no longer be able to say this:
Never mind what makes our modal opinions count as knowledge; how do we come by the modal opinions that we do in fact hold? In the mathematical case, the answer is that we come by our opinions largely by reasoning from general principles that we already accept; sometimes in a precise and rigorous way, sometimes in a more informal fashion, as when we reject arbitrary-seeming limits on the plenitude of the mathematical universe. Suppose the answer in the modal case is similar. I think our everyday modal opinions are, in large measure, consequences of a principle of recombination. (On the Plurality of Worlds, p. 113)
So what we learn from this exercise, maybe, is that the above, seemingly casual, argument is more important to Lewis than one might have thought.
He needs to isolate a non-analytic general principle behind many of our common sense modal opinions; otherwise, the business of unifying and systematizing them will not get very far.