Here is an interesting problem for Lewis. Lewis says that sets are necessary beings, and hence count as existing in all worlds. Very plausibly then:
- If A is a set and w1 and w2 are worlds, then A in w2 is a counterpart of A in w1.
After all, if identity isn’t good enough for being a counterpart, nothing is. Note that (1) does not say that A at w2 is the only counterpart of A in w1. To handle some identical twin scenarios, Lewis may need to allow a world to have more than one counterpart of an object.
Let α be Aristotle. Let A = {α} be the singleton of α. Lewis is now committed to the truth of:
- Possibly α is not a member of A.
For Lewis’s criterion for whether F(a, b) is possible is whether there is a world w with counterparts a′ and b′ of a and b respectively such that F(a′,b′) holds at w. Let w be a non-actual world where there is a counterpart β of Aristotle. Since individuals are world-bound, β ≠ α. Moreover set membership is necessary, so:
- β is not a member of A at w.
Since β is a counterpart of α and A is a counterpart of A by (1), it follows that (2) is true. But (2) seems clearly wrong: it is impossible for Aristotle not to be a member of A.
Here’s what seems to me to be the best way out for Lewis: Require pairwise counterparts rather than individual counterparts (in fact, I vaguely remember that Lewis may do that somewhere) for possibility claims involving two objects. Thus that β is not a member of A and β and A are individually counterparts of α and A isn’t enough to make it be that possibly α is not a member of A. One would need β and A to be pairwise counterparts of α and A. But perhaps they’re not. Perhaps, rather, it is β and B = {β} that are pairwise counterparts of α and A. However, this greatly complicates the counterpart relation as well as Lewis’s identification of properties with sets.
