One benefit of admitting the existence of possibilia, says Lewis, is the identification of properties with sets of possibilia. I've always been confused as to why this is meant to be better than the actualist saying that proeprties are sets of their instances.
Lewis says, "The usual objection to taking properties as sets is that different properties may happen to be coextensive. . . the property of having a heart is different from the property of having a kidney, since there could have been a creature with a heart but no kidneys." And this is usually the reason I'm given when I ask this question.
But the 'since' is no good! The fact that there could have been a creature with a heart but no kidneys shows only that the property of being a renate might not have been the property of being a cordate. That only tells us that the properties are actually distinct, and hence that the properties aren't actually identical to their actual instances, if we accept the (necessity of) the necessity of non-distinctness. But Lewis *doesn't* accept that, due to his acceptance of counterpart theory.
Assuming contingent identity in general is not incoherent, what would be wrong with someone holding that being a cordate *is* identical to being a renate, but only contingently so? On this view, something might have had being a cordate and lacked being a renate because the actually identical properties might have been distinct. Whenever Lewis holds that two distinct properties are accidentally coextensive this theorist holds that they are contingently identical; properties that Lewis identifies, this theorist claims to be necessarily identical. Would anything go wrong with this?
You might object to the proposal on the grounds that, even if contingent identity in general is okay, it's not okay for sets to be contingently identical. Why? Because sets have their members essentially and because the axiom of extensionality is necessary. Those two claims entail that identical sets are necessarily identical. (Proof: if S and S* are identical they share their members. Given the essentiality of membership they share their members in all worlds. So given the necessity of extensionality they are identical in all worlds.) But I don't find this that convincing from the perspective of the counterpart theorist. I think the counterpart theorist should hold that whether sets have their members essentially is a context-sensitive matter, just as it is a context-sensitive matter whether or not I am essentially human. When we specify a set extensionally - 'the set of a, b, and c' - it's natural to suppose we invoke a context whereby nothing gets to be a counterpart of that set unless its members are the counterparts of a, b and c (what we say if one of a, b or c has multiple counterparts at a world is going to get tricky). But when we specify a set intensionally - 'the set of the Fs' - it's natural to suppose that the counterpart of this set at a world is the set of the things at that world that are F. It doesn't seem objectionable to me, then, to say that 'the set of the cordates' and 'the set of the renates' are contingently identical which, on the current proposal, is just what it is for the proeprties being a cordate and being a renate to be contingently identical. Being such that 2+2=4 and being such that everything is self-identical, on the other hand, will be necessarily identical, because at every world the set of things that are such that 2+2=4 is identical to the set of things that are such thateverything is self-identical - namely, it is the set containing everything at that world.
I'd be interested to hear reasons for not going this way.