Extreme Haecceitism
Geoff at Too Much Text points out that the implausible hyper-essentialism implied by Kripke's account of rigidity can be avoided by adopting radical anti-essentialism, the view that there are no non-trivial (qualitative) essential properties at all. On this view, even though there is a precise boundary between a thing's essential and non-essential properties, the boundary is not very mysterious because it classifies virtually all properties as non-essential.
Radical anti-essentialism implies that individual possibilities are more fine-grained than qualitatively individuated possible worlds. That is, once a possible world is completely described in qualitative terms, it is still open which individual, if any, I am at that world: Consider a world where there is a unicorn, a poached egg, and nothing else. By assumption, I could have had the property of being a unicorn in a world in which there is only a unicorn and a poached egg; I could also have had the property of being the poached egg in such a world. So the qualitative description of this world cannot settle what I am.
The most obvious way to express radical anti-essentialism in the possible worlds framework is therefore what Lewis calls extreme haecceitism (Plurality §4.4): Every possible world has indefinitely many qualitative duplicates differing only in which individuals play which qualitative roles. Alternatively, one can let a single world do the work of all its qualitative duplicates if one accepts 1) counterpart theory, 2) a minor revision in the possible worlds analysis of de re predications, and 3) very relaxed restrictions on counterparts. The minor revision is to regard "I could have been a poached egg" as true iff at some world, at least one of me -- that is, of my counterparts -- is a poached egg. (This is what Lewis offers to satisfy haecceitist intuitions.)
Geoff is right: extreme haeccesitism is a very attractive position. Well, it gets very attractive once we add an explanation why in some contexts, "I could have been a poached egg" seems false: in those contexts the various possibilities of me being a poached egg are ignored. With this addition, extreme haecceitism shares most of the virtues of counterpart theory. In particular, it can explain why essentialist predications are so vague and context-dependent: because it is vague and context-dependent which possibilities are ignored. It would be nice if extreme haecceitism shared all the virtues of counterpart theory, but unfortunately it doesn't: Unlike counterpart theory, the most natural reading of extreme haecceitism entails the necessity of identity. That's unfortunate because I would have liked to say that any theory that entails the necessity of identity has wildly implausible consequences such as hyper-essentialism. Now it seems that there is a theory of this kind that only has mildly implausible consequences such as that I could have been a poached egg. I have to think more about this.
Extreme haecceitism does not entail the necessity of identity if some identicals are contingently *self*-identical. For if (1) holds,
(1) EyEx(x=y & ~[](x=x & y=y))
then from (2)
2) AxAy{x=y -> ([](x=x & y=y) <-> [](x=y))}
it follows that not all identicals are necessarily identical:
3) ExEy(x=y & ~[](x=y))
Strongly reflexive identity precludes such a domain. But weakly reflexive identity permits it. Identity is weakly reflexive if Ax(x=x <-> Ey(x=y)) holds but Ax(x=x) and AxEy(x=y) do not.
A domain in which identity is weakly reflexive is described by Chris Mortensen, who writes:
"It is possible to produce ... structures in which the domain is divided into two disjoint classes, those for which n = n but not also n =/ n, and those for which n =/ n but not also n = n; which gives a universe in which some things are self-identical and some (other) things are non-self-identical. Again, a different approach might take self-identity as the criterion of existence. This is not uncommon; recall the definition of the null set = df {x: x =/ x). This gives the possibility of a universe in which nothing is self-identical and everything is non-self-identical.
Indeed, this universe seems to be consistent. At least, it is consistent if it is consistent that nothing exists, which has seemed a desirable proposition to some. ("Anything is Possible", p. 330)