The only x and y principle states that the relation between "x=y" only depends on facts about x & y. So if we say something like "Bob is this particular human being" then this relationship is only determined by facts about "Bob" and facts about "this particular human being". Many people seem to say that violating this principle is a problem for personal identity issues. But I really don't see how this is an issue. For instance, facts about "Bob" are contingent on facts about "Bob's mother" who named him. What's the main problem about violating this?
The 'only x and y principle' was introduced by Harold Noonan. Whether this is the best name for the principle is questionable but here is what Noonan says :
A plausible principle governing identity is that whether a later individual, y, is identical with an earlier individual, x, cannot ever merely depend on whether there are, at the later time, any better candidates than y around for identity with x. I shall call this principle 'the only x and y principle'. ( H. W. Noonan, 'Wiggins, Artefact Identity and 'Best Candidate' Theories', Analysis, Vol. 45, No. 1 (Jan., 1985), p.4.)
Noonan here and elsewhere ('The Only X and Y Principle', Analysis, Vol. 45, No. 2 (Mar., 1985), pp. 79-83) has complex arguments for the principle but I think it can be explained by means of a fairly straightforward example.
Take X at time t. Over time, X will change : at time t+1 X might have significantly different features from X at time t. Now suppose that at time t+1, there is a Y which has more resembling features to X at time t than X himself or herself has at time t+1. The claim might be made that Y at time t+1 is the closest continuator of X at time t and therefore the 'best candidate' to be X at time t+1.
X at time t and Y at time t+1 stand in no 'ancestral' relationship. There is no causal relationship between them, only the fact of resemblance. There clearly is resemblance, similarity, between X at t and Y at t+1 but according to the principle, X is not the same person over time as Y because X and Y have independent histories and there was no prior time at which they were identical. Whether they are identical or not depends purely on facts about each of them - facts about their histories over time. Their histories are distinct; and they are not identical at t+1 merely in respect of the contingent fact that at t+1, Y has more resembling features to X at time t than X himself or herself has at time t+1.