Essentialists take seriously the idea of de re modality, and claim that "the modal properties of an individual (properties such as being essentially F or possibly G) are had independently of the way in which the individual is referred to." (Della Rocca, p.226.)
I once outlined a couple of Quinean anti-essentialist arguments in this old post:
Quine points out that a single object X can be equally well specified by either of the following two descriptions:
(1) The number of planets in our solar system
(2) 7 + 2
Now, he asks, is it a necessary truth about X that it is a number greater than 7? Well, it depends which specification you use. Of course [7 + 2] is necessarily greater than 7; but [the number of planets in our solar system] surely is not. So, it seems, we can't really say anything about X objectively, i.e. independently of how we specify it.
Such concerns may be countered by noting that description (2), but not (1), is a rigid designator, referring to the same object X -- the number 9 -- in all possible worlds. There are possible worlds in which description (1) would be satisfied by some object other than X, but this does nothing to show that the object X itself could fail to be greater than 7.
But note that rigid designation presupposes trans-world identity. You can't pick out the same object in all possible worlds unless there's some criterion to determine which other-worldly objects are the same as each other. It requires that possible worlds come with objects' identities "built in", so to speak. (Kripke proposes that we simply stipulate that we're talking about the possibility in which this very man wins the election, or whatever.) Well, perhaps that's too strict. Really all we require is that there be an objective answer to the question of "which object is X?" in any given possible world. The answer needn't be explicitly "built in", so long as we can extract it, say with an appropriate counterpart theory. If the counterpart relation is objective and determinate, then the essentialist could obtain something close enough to "rigid designation" simply by adopting that intension which picks out the counterpart of X in each world where one exists.
What the anti-essentialist requires, then, is a relativistic counterpart theory. That would mean that there's no uniquely correct answer to "which object is [the counterpart of] X?" in a possible world W. Rather, it will depend on which actual properties of X are most salient in the present context. The counterpart of X in W will be whatever W-object is most relevantly similar to the actual X, but what's "relevant" can vary from context to context. So one cannot hope to rigidly designate X any longer, since there's no objective fact as to which W-object is X. (Sometimes you'll want Y as the counterpart, and other times, Z.)
On this view (recently mentioned in passing here), there are no haecceities, or deep metaphysical facts about identity. Of course it's trivially true that each object (in world w at time t) is self-identical. But there are no objective facts about trans-world identities (nor presumably cross-temporal ones either -- compare Parfitian reductionism about personal identity). There are merely distributions of property-clusters across space and time, and there is no "further fact" about whether two such clusters are really the "same" object. Once we settle the qualitative facts, there is no further work to do. If it's agreed that the Twin Earth scenario is possible, no deep question remains as to whether the watery stuff in the lakes and rivers is really water. The difference is merely semantic.
We can still talk about modal properties, but they really belong to the words, not the objects. To say that water is necessary H2O is merely to say that our terms 'water' and 'H2O' have the same (secondary) intension. It's a metalinguistic claim, not a metaphysical one. To say that I'm "essentially human" is merely to say that anything non-human is excluded from the intension of 'RC'. It's not really saying anything deep about me, or my nature across other possible worlds. For the anti-essentialist, there's nothing deep to be said. (That's not to deny that there can be practical reasons for adopting some intensions rather than others.)
Suppose the world could have been such as to contain nothing but a pair of qualitatively indiscernible dice. We can give a full description of them: red with white pips, a certain size, etc. Are they the same dice as the ones on my desk? Or a pair that happen to actually be in Las Vegas? Or are they not to be identified with any actual dice at all? The anti-essentialist suggests that these are empty questions. The qualitative description gave you the possibility. There aren't any further "identity facts" to settle. There aren't different possibilities corresponding to each possible identity of the dice. No, there's just the one scenario being described, and we've already said what it contains.
I find that part of the story pretty plausible. Here's a more radical step: the dice roll, and one lands on '3', the other on '4'. Might it instead have happened that the former landed on '4', and the other on '3'? Is that a different possibility? It could be if this time the '4' lands first, or in a different position, or otherwise involves a different distribution of qualitative properties through space and time. But suppose all that remains the same. What we're considering is a switch in identities alone, with all else held fixed. If you think identity is a "further fact", then such an haecceital "switch" should be possible. But I think I want to deny this, and again claim that there's just the one scenario here, so that once the full qualitative description is given, there's nothing more left open.
After all, if you allow the two dice to switch identities without any outward symptom of the change, where do you draw the line? Couldn't just one of them switch identities with another non-existent die? Or perhaps the other could become the Las Vegas die? There would be an infinity of qualitatively indiscernible possible worlds, and that seems a tad excessive.
Are there any serious disadvantages to the anti-essentialist, anti-haecceitist view? One might worry that it returns the wrong probability verdicts: we should think it twice as likely to roll a 3 and 4 than two 4s, presumably because there are two ways the former can occur. But you don't need haecceities to recognize that. When you look at the full range of qualitative possibilities, including some where the 3 lands x seconds before the 4, and vice versa, there may be indefinitely many qualitatively discernible possibilities. If we look at the ratios of resulting frequencies, we'd presumably find that twice as many of these scenarios involve a 3 and a 4, as opposed to two 4s. No haecceities needed.
P.S. I think this view might be equivalent to a super-essentialist view which sees all -- even extrinsic or relational -- properties as "essential" to an object. We could effectively deny that anything in other possible worlds is identical to the actual me, just like, strictly speaking, my future temporal parts are non-identical to my past temporal parts. We have no essence that endures through time, nor across modal space either. You can construct four-dimensional spacetime "worms", and if we add other possible worlds into the mix then that could give a fifth dimension. But there's something a little bit arbitrary about the resulting entities. Some are more gerrymandered than others, and we'll generally find it more useful to talk about the less blatantly gerrymandered ones. But we might deny that these are wholly natural divisions of reality in the first place. We might have at least some discretion to divide up the worlds into 5-D "objects" as we please. And of course when you draw the lines yourself, it doesn't mean a lot when you later note that they never cross certain boundaries. You're merely commenting on your own classificatory habits, not the deep structure of reality-in-itself.
Reference: Michael Della Rocca (2002) 'Essentialism versus Essentialism', in Gendler and Hawthorne (eds.) Conceivability and Possibility.
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