This is a reconstruction of the argument between Kripke and Quine about the connection between rigid designators and essentialism.
(i) A term is a rigid designator if it picks out the same property, n in all possible world-states in which the object o containing that property exists.
(ii) A property P is essential to an object o when the claim it is necessary that if o exists then o is P, is true
(iii) If n is a rigid designator of o, and F is a predicate expressing the property P, then the claim that P is an essential property of o is the equivalent to the claim it is necessary that if n exists, then n is F.
(iv) There will be some terms t that refer to o which make the sentence it is necessary that if t exists, then t is F false.
(v) There is no principled, non- arbitrary way of selecting, for an arbitrary object o and property P, what sort of term t should be used to underwrite claims to the effect that o did, or did not, have P essentially.
(vi) So, objects have or lack properties essentially only relative to ways of describing them.
I think that there is a problem with premise (iv). It is necessary that if I, Natalie exist, then I am a human. This is true because it is the case that in all possible worlds, to be considered Natalie, it is a necessary property to be human. I could also say that it is necessary that if I exist, then I have a dog named Arthur. This is false because it could be the case that in some possible world I do not have a dog, or I have a dog and his name is Bob, etc. This tells me that the property of having a dog named Arthur is not an essential property of being Natalie. So then having a dog named Arthur is not a rigid designator of Natalie.
Quine’s example gives two separate properties to an individual i, that make the sentence it is necessary that if t exists, then t is F false. It goes like this:
Let i be some individual who is both a brilliant mathematician and a champion cyclist, and suppose that the world’s greatest mathematician and the world’s greatest cyclist both designate i. Then (a) is false.
(a) it is necessary that: if the world’s greatest mathematician exists, then the world’s greatest mathematician is two-legged.
So, premise (iv) seems correct in claiming that there are some instances where the sentence it is necessary that if t exists, then t is F is false.
There seems to be something fishy with this example; if i is an individual that is rigidly designated by the properties brilliant mathematician and champion cyclist then wouldn’t (a) be true because if the world’s greatest mathematician is a champion cyclist, then it follows that the world’s greatest mathematician is two legged. So, I think that Quine’s example doesn’t necessarily prove premise (iv).