Here I want to motivate and defend two Parfitian theses. The first is anti-haecceitism or 'reductionism' about identity: the qualitative facts exhaust the facts, and in particular, there remains no "further fact" about the identities of things. We could give a complete description of the world without using identity-talk at all. Talk of identity over time is just a convenient shorthand for talking about various kinds of continuity and counterfactual dependence. The second thesis, then, is that when it comes to the persistence or identity of persons across time, the kind of continuity worth caring about is psychological continuity.
Consider a world (w1) containing two qualitatively identical soccer balls. Now consider the distinct possibility (w2) that just one such ball ever existed. Once we specify all the qualitative details -- e.g. the size, shape, and mass of the ball -- is there any further matter that we remain ignorant of? In particular, could we sensibly wonder about the identity of the ball, e.g. whether it is "really" either of the two balls that would have existed in possibility w1? Presumably not. It is difficult to comprehend what in the world a mere difference of identity would consist in. One proposal is that there are 'haecceities' -- properties such as being ball_1 or being ball_2, which float free from the qualitative natures of the objects. But then it is hard to see what in this theory precludes such seemingly nonsensical behaviour as objects switching haecceities from one moment to the next -- or even instantiating multiple such properties at once.
What's more, once we shift our attention from trans-world to trans-time identity, we can construct fission cases that seem to pretty conclusively establish that identity across time is not intrinsic. In short: Fred would survive if just his left brain hemisphere were successfully transplanted into a new body ('Lefty'), and likewise if just his right hemisphere were transplanted into a different body ('Righty'); but if both Lefty and Righty survive, then - since they are distinct from each other - they cannot both be identical to Fred. Hence, whether Fred (at time t1) and Lefty (at t2) are the "same person" does not depend just on the intrinsic properties of the respective person-stages; it also depends on whether there is any other stage (e.g. Righty) with an equal or better claim to being Fred's closest continuant at t2.
So the identity facts are not "built into" the objects themselves. They are better understood as merely conventional: we may say that "Lefty is Fred", when Lefty is a sufficiently close continuant of Fred and there are no real competitors for the title; but we aren't tracking any "deep further fact" about the world when we do so. If Righty exists as well, then we may refrain from bestowing the title of '[closest continuant of] Fred' on either of the equally-eligible candidates, but of course Lefty himself is not any different for lacking the title. It is merely a difference in how we talk, not a difference that Lefty need regard as having any real significance.
This deflationary, reductionist view may also be supported by examples like the Ship of Theseus, or the following example (adapted from Parfit): Suppose Jim has fond memories of a now-defunct social club that he founded as a youth. He decides to start a similar club, with the same name and membership rules as his old club, but new members. Once we have all these qualitative details about the connections between the two clubs, it'd clearly be vacuous to ask whether or not they are really "the same" club. It's not as though there are two open possibilities here. We know what the situation is; the only remaining question is how we choose to describe it. (See also: 'Arbitrary Persistence'.)
The counter-intuitive step comes when we apply this same reductionist understanding to questions of personal identity. When faced with teletransporter cases and the like, we're inclined to think that there must be some deep further fact about whether the person that emerges on the other side is really the same conscious experiencer as the person who went in, or a mere replica. (I'm tempted to phrase the intuitive worry in terms of whether the same 'stream of consciousness' continues on, or whether a new 'stream' begins where the last one left off. But that can't be quite right: we ordinarily think that disconnected streams of consciousness -- e.g. before and after sleeping -- can be experienced by the same enduring person or 'subject of experiences'.)
This way of thinking seems endemic to our self-perception: we don't just think that there will exist a future stage, related to our present stage in various ways, who will experience such-and-such. We anticipate experiencing that future ourselves. And so puzzle cases like teletransporters and fission cases strike us as puzzling precisely because we assume that there is an important first-personal difference between whether those futures are experienced by us or instead by other minds that merely happen to be very similar to ours. Those seem like distinct possibilities, in a way that alternative answers about the identities of inanimate objects (like the Ship of Theseus, or Jim's social club) do not.
I feel the force of this intuition, but I think we should ultimately reject it as a mere psychological confusion. The previously mentioned theoretical reasons for rejecting haecceitism in general strike me as extremely powerful, and they would seem to apply no less strongly when it comes to the identities of mental entities. (See also Velleman on the illusion of endurance.) But I'd be interested to hear how others respond to this dilemma.
[Aside: Parfit himself seems to regard reductionism as merely contingently true. I'm inclined to the stronger view that non-reductionism about identity isn't even a coherent possibility.]
II. Psychological Continuity
Given that there are no "deep further facts" about identity for our concerns to track, what should we care about in survival? As Parfit argues, one thing that fission cases show is that we shouldn't care about conventional identity ascriptions, since identity depends on extrinsic facts, whereas Fred's attitude towards Lefty should depend only on their intrinsic features and relations. Since Fred would regard himself as 'surviving' if just Lefty (or Righty) survived, he should regard the outcome where both Lefty and Righty survive as a kind of "double survival", even if we can't strictly speaking identify him with either survivor. [See also 'Fissioning in Prospect and Retrospect', where I argue that fissioning followed by the painless death of Righty is about as good as ordinary survival.]
Okay, so we should care about some (multiply instantiable) relation of similarity/continuity. But which one? The possibility of surviving as a 'brain in a vat' seems to rule out considering one's body as essential. So the serious contenders seem to be:
(i) the physical continuity of your particular brain (whatever general psychological qualities it might give rise to), or else...
(ii) the continuity of your general psychological qualities (memories, values, personality, etc.), whatever physical entity these might be based in.
The first option might seem intuitive to the extent that one retains the Cartesian picture of an enduring self. But once we accept reductionism, the second option seems to make a lot more sense. I care about my memories, values, etc., and I want to see my intentions carried out and my projects brought to fruition. It doesn't make any difference to me whether the Richard-like mind that takes care of my future business is a mind that's grounded in a brain physically continuous with my current brain, or whether future-Richard's brain is instead reconstructed from all new atoms on the other end of a teletransporter 'journey'. I don't necessarily have any argument to offer to someone who finds that they continue to intrinsically care about the persistence of their particular brain even after accepting reductionism; such a response just seems inexplicable to me, so I assume that responses favouring the psychological view will be much more common.
P.S. One might not even need to accept reductionism in order to be drawn to the view that psychological continuity is what matters in survival. Locke pointed out that however seriously we entertain the hypothesis that we share a bare "soul" with Hector of Troy, so long as we are not thereby supposed to inherit any of his memories or other psychological qualities, we find ourselves unable to really consider ourselves the same person as Hector in any sense that matters. On the other hand, if a Cobbler inherits the memories and personality of a long-dead Prince (completely overriding any trace of the Cobbler's mind), for everyone involved it is as if the Cobbler has been killed and the Prince resurrected in his place -- no matter whether there has been a change of haecceity (or identity of the immortal "soul" that we imagine to inhabit the body).