While I have my doubts about the claim that all identity is sortal-relative (i.e., that things are never merely identical to each other but always only are the same something-or-other), it always astounds me how bad some of the arguments against relative-identity theorists are. One very popular argument that strikes me as bad is the following (with variations).
Suppose you have two individuals, a and b; and these two individuals are each of a certain kind, K. On this supposition, a is a K, b is a K, and a is not the same K as b. If a is a K, it's certain that a can't be a different K from a. That means that a and b are distinguishable (discernible, to use the common term in this type of situation). If a and b are discernible, however, they cannot be the very same K*, because they cannot be the very same anything. So if a and b are each K's and K*'s, they can only be the same K when they are the same K* (whatever K* may be). Thus identity is not sortal-relative.
There are a number of obvious problems with this type of argument. But the most obvious is that the relative-identity theorist needn't -- and, one would imagine, wouldn't -- accept the claim that if if a and b are of a different sort that a and b can't be the same sort (when the 'sort' is different). If I say, "The blue of this chair is the same color as the blue of that wall," it's not an adequate argument against taking this as a sortal-relative identity that the two are obviously different colors because one is a chair-color and the other is a wall-color. The relative-identity theorist will just say, and probably rightly, that you have stopped giving serious arguments and are now just being silly. But that's precisely what we have here. Let "The blue of the chair" be a, and "The blue of the wall" be b. Now, "The blue of the chair" is obviously not absolutely identical to "The blue of the wall"; obviously, as the silly objector said, one is the color of the chair and the other is the color of the wall. So they differ. But they are the same color, ex hypothesi. Now, we know that "The blue of the chair" can't differ from "The blue of the chair". That means there's a way in which "The blue of the chair" and "The blue of the wall" can be distinguished -- one can differ from "The blue of the chair" and the other can't. But, says the objector, this means that they can't be the same anything as each other. And, of course, if this is taken broadly, the objector is obviously wrong: as we've already said, they are the same color. But if it's taken in a more narrow sense, the objector is just begging the question: the only way it follows from the premises is if the only sort of identity is not sortal-relative.
A common counter-response to this is to note that we make arguments of the following form all the time:
a is P
a is the same K as b
∴ b is P.
And this is true. If the relative-identity theorist is right, however, inferences of this form are not (as written here) valid; sometimes the inference will give you the right conclusion, and sometimes not. When it does, it will have to be because it is enthymematic. And, it would seem, we have no account of just when these inferences work and when they don't. But I don't see that there's any great mystery here. Obviously, the inferences will work as enthymemes when the account for why a is P and the account for why b is P have something relevant in common. The most obvious such case would be when a is P because it is K, the very same K that b is. In such a case, b would obviously be P, because it would be P for the very same reason a is, by the very nature of the thing.
So this line of thought seems to me to be a dead-end in this dispute. There are other, more interesting arguments, of course. And it's possible that I'm just missing something obvious. But the argument just doesn't seem workable.
