I agree with most of what Parfit says about personal identity. But I'm skeptical of one claim he makes. He suggests that he might have been wrong -- that although in fact reductionism is true, and personal identity is not a further fact over and above physical and psychological continuity, things could have turned out otherwise.
Parfit thinks that persons could have been irreducible Cartesian Pure Egos, spiritual substances whose continued existence is all or nothing. Possible evidence for this would be provided by apparent reincarnation, i.e. if someone had apparently accurate memories/knowledge about the life of someone long dead, which they could not have learned by natural means. This would be evidence that we are spiritual substances. And if we found that we could not induce intermediate degrees of psychological continuity -- say that brain damage either left you completely unaltered or a completely different person, with nothing in between -- then that might be evidence that the continued existence of the ego is all-or-nothing, and does not admit of vagueness or borderline cases.
Even granting all this, is it non-reductionism? Or does it instead show that rather than just being reducible to physical stuff and psychological continuity, our personal identity may instead have had a slightly different (soul-stuff based) reduction basis? I'm not seeing any significant difference here. Suppose that (1) our psychological traits inhere in a spiritual substance or 'ego' that exists independently of, but interacts with, the physical; and (2) these psychological traits are not susceptible to incremental change or manipulation -- their persistence is "all or nothing". What then? It does not seem to follow that (3) the ego endures rather than perdures, nor (4) each ego has a haecceity or essentially unique property or 'thisness' which fixes its identity as a further fact over and above the qualitative facts.
If identity is a 'deep further fact', one of the fundamental base facts from which all other truths may be derived, then it should vary independently from, rather than supervening upon, the other fundamental facts. (Shouldn't it?) There must be two possible worlds which are exactly similar but which vary in the identity facts -- perhaps due to switched haecceities. That seems a bit nutty, which is why we should be reductionists about identity in general. But note that adding spiritual substances of limited qualitative repertoire into the world doesn't seem to change this in the slightest. We can still think it an empty question whether the corresponding egos in two possible worlds are numerically identical or merely exactly similar. We can still think that the identities of the various egos in the world reduce to qualitative facts about what the world is like (there is some soul-stuff over here, housing such-and-such a personality).
Consider (apparent) reincarnation. What makes your ego "one and the same" as that of a Japanese woman who lived a thousand years ago? Presumably, this identity fact is reducible to the psychological continuity and counterfactual dependence that holds between you. That is, you have her memories etc., and this involves a kind of robust tracking such that if she had lived differently then you would now have appropriately different memories (etc.) in their place. It is this which inspires us to combine the various temporal parts into a single, unified ego (without it seeming objectionably gerrymandered). There is no 'further fact' in the sense of haecceities or other special identity facts. We have simply imagined a scenario where there are some further qualitative facts which go beyond the sorts of substances and connections that are actually found in our world. I see nothing here for the non-reductionist to get excited about. Am I missing something?