Here is an interesting line of thought.
Consider these two questions:
- What is it for x and y to be identical?
- What necessary and sufficient conditions can be given for a person to be identical with a person?
- What is it for an action to be permissible?
- What necessary and sufficient conditions can be given for a killing of a person to be permissible?
One way to notice the difference I'd like to highlight between (3) and (4) is to consider the kinds of answers one might get. For instance, on a fairly standard natural law theory, the answer to (3) is that an action is permissible if and only if it does not conflict with one's nature, while the answer to (4) is, perhaps, that the killing be a proportionate act of justice or self-defense and not otherwise impermissible. On a Kantian theory, the answer to (3) is that an action is permissible if and only if it treats no person as a mere means. But the answer to (4) may be exactly like the natural law answer. On a divine command theory, the answer to (3) is that what makes an action permissible is its not being forbidden by God, but again the answer to (4) might just like on the other theories.
Observe, thus, what we are very unlikely to get from standard answers to (4): we are very unlikely to find out what the permissibility of a killing consists in. If we want an answer to that question, the natural lawyer will say that the permissibility of a killing consists in its comformability to our nature, the Kantian that it consists in its not treating anyone as a mere means, and the divine command theorists that it consists in its not being forbidden by God. None of these answers will answer the applied ethics question we want, as these answers are at too high a level of generality: we can replace "killing" by any other action type, and they remain applicable.
Now go back to the metaphysics. I have no idea how to answer (1). It's hard to think of anything more fundamental than identity to answer it in terms of. But I think the analogy with the ethics question suggests this. If we can get a substantive answer to (2), it's not going to be an answer to the question of what identity of persons consists in. It's simply going to be an answer as to what interesting necessary and sufficient conditions for identity are in the special case where it is given that the relata are persons.
If we want to know what the identity of persons (or at least finite persons) consists in, the correct answer will not be so nformative: what makes person a be identical with person b is that (i) a is a person, (ii) b is a person, and (iii) Iab, where "Iab" is a stand-in for whatever identity in general consists in. I have no idea what "Iab" will say, but I know that it won't say anything about memories, gradual replacement of cells, etc. For, "Iab" is the general account of identity, and that is beyond the details of particular kinds of beings.
This line of thought is attractive, but resistable. One might instead insist that identity is something different in different types of beings. (This may or may not involve the further step of accepting a relative identity theory.) Here, the "types" could be categories—substances, accidents, relations, etc. Or they could be kinds—dogs, persons, photons, electromagnetic fields, etc. I prefer the first option, but what I say applies in both cases. One way to flesh out such views is with a theory of analogy. There is no one thing that identity consists in: there is a relation between a substance and substance, and a relation between an accident and an accident, and so on, and these are all analogous (maybe one of them is focal?). Or if one prefers the second type of "type", one might say: there is a relation between a dog and a dog, and a relation between a person and a person, and so on, and these are all analogous.
On this kind of view, we might well have something substantive to say about what identity between, say, a person and a person consists in, which does not reduce simply to saying: this is a person, and this is a person, and this equals this. I think very naturally such a view will call for a typed logic. (Query to self: In a typed logic of this sort, should there be a quantification over types of quantification?)
I wonder if the argument of my recent AJP paper on diachronic identity is vulnerable to this sort of view. Quite possibly. But I still think the reduction of diachronic to synchronic identity that I perform in that paper is plausible.