According to the identity of indiscernibles, as a matter of necessity, no two things can share all their properties. If a and b share all their properties then they are not two things, but one: they are numerically identical. You can’t have qualitative identity without thereby having numerical identity.
Why think the identity of indiscernibles is true? Well, suppose it’s false; in that case there is a possible world containing two entities, a and b, which share all their properties. And the uncomfortable thought is meant to be that in that world there is nothing that makes these entities distinct.
This thought proceeds from the idea that, for every object o, something grounds the identity of o: there is a metaphysical explanation for o’s being that very object, and not some other thing. What could do this grounding other than some subset of o’s properties? But if o is the thing it is in virtue of having some subset of its properties, then any thing which had all and only o’s properties would have its identity determined in the same way as o, and hence would be numerically identical to o.
That, I take it, is the intuitive thought behind the identity of indiscernibles. Any defender of that thought has to say something about Max Black’s world consisting solely of the two homogenous iron spheres Castor and Pollux. One thing to say is that the objects have different haecceitistic properties. Castor is distinct from Pollux because one has the property being identical to Castor and the other the property being identical to Pollux. Something seems unsatisfying about that.
Firstly, you might not like the admission of such a ‘property’ into your ontology. Our grasp of what properties are seems to be tied to qualitative ways for things to be – such as being red or being round etc; the introduction of mere haecceitistic properties might be thought to have stretched the concept of property beyond breaking point.
But suppose you accept the existence of properties of the form being identical to x; there is a deeper problem. We are told that Castor and Pollux are distinct because Castor has the property being identical to Castor and Pollux has the property being identical to Pollux. But that only serves to ground the distinctness of Castor and Pollux if these properties are themselves distinct. If ‘the property being identical to Castor’ and ‘the property being identical to Pollux’ are two names for the same property then the fact that Castor has the former property and Pollux the latter doesn’t serve to ground their distinctness. So we need to ensure that these haecceitistic properties are distinct if their admission into our ontology is going to help us with Black’s example. But then, by the same reasoning that led us to the identity of indiscernibles in the first place, there must be something that grounds the distinctness of these properties. Now what do we usually think are the individuation conditions for properties? One might try: P and Q are the same property iff they make the same contribution to the qualitative nature of their bearer. But of course, this will only do for qualitative properties; applying it to mere haecceitistic properties would make them all identical, since mere haecceitistic properties don’t make any contribution to the qualitative nature of their bearers (that’s precisely why it rankles to call them ‘properties’). So for mere haecceitistic properties it seems we should have instead: P and Q are the same haecceitistic property iff they belong to the same thing.
But if that’s right then being identical to Castor is distinct from being identical to Pollux in virtue of the bearer of the former being distinct from the bearer of the latter. In which case we can hardly appeal to the distinctness of the properties to ground the distinctness of the bearers.
One who invokes mere haecceitistic properties to deal with Black’s world wants to hold that [Castor is distinct from Pollux] in virtue of the distinctness of being identical to Castor and being identical to Pollux; but it seems that they should be committed to [being identical to Castor is distinct from being identical to Pollux] holding true in virtue of the distinctness of Castor and Pollux. But these ‘in virtue of’ claims can’t both be true, since in-virtue-of is asymmetric. So the introduction of mere haecceitistic properties doesn’t seem to help: to ground the distinctness of Castor and Pollux the distinctness of the properties would apparently need to be taken as brute; and if we’re prepared to do that, why not just accept the distinctness of Castor and Pollux as brute in the first place?