Consider split brain patients, who have their two hemispheres surgically disconnected. The evidence suggests that this gives rise to two independent spheres of awareness, each with access to only one half of the visual field, and controlling one half of the body. Parfit offers a simplified account (p.245):
One of these people is shown a wide screen, whose left half is red and right half is blue. On each half in a darker shade are the words, 'How many colours do you see?' With both hands the person writes, 'Only one'. The words are now changed to read: 'Which is the only colour that you can see?' With one of his hands the person writes 'Red', with the other he writes 'Blue'.
Now imagine technology advances to a point where we could temporarily split our brains (e.g. by anesthetizing the corpus callosum):
When I disconnect my hemispheres, my stream of consciousness divides. But this division is not something that I experience. Each of my two streams of consciousness seems to have been straightforwardly continuous with my one stream of consciousness up to the moment of division. The only changes in each stream are the disappearance of half my visual field and the loss of sensation in, and control over, one of my arms.
[Ten minutes later...] I am about to reunite my mind. What should I, in each stream, expect? Simply that I shall suddenly seem to remember just having worked at two calculations, in working at each of which I was not aware of working at the other. This, I suggest, we can imagine. And, if my mind had been divided, my apparent memories would be correct. [Parfit, p.247]
How are we to explain the temporary disunity of consciousness? It is implausible to suggest that any new people came into existence for just those ten minutes. Rather, we should say that the one person had a (temporarily) divided mind, with two separate spheres of awareness, neither of which was aware of the other. It follows that a person is not just a 'subject of experiences', and cannot explain the 'unity of consciousness'.
Now suppose that your body and your identical twin's brain have been fatally injured. If your brain could be transplanted into his body, then you would survive. In fact, one hemisphere is enough - people have survived strokes or injuries that destroyed just one of their hemispheres. So imagine that one half of your brain is transplanted successfully, and the other half is destroyed. We must say in such a case that you survive in your twin's body.
But what if the other half is not destroyed? Suppose you have identical triplets. We split your brain in half and transplant each into different bodies. This seems to be an even better result for you than the previous scenario. It seems twice as good. However, it is one where your personal identity is lost.
We clearly have two surviving people, not just one. You cannot be both of them. They are entirely independent now - each with all the characteristics of an individual person. It is not like the 'divided mind' case when both streams soon recombined. Here the separation is permanent, so we cannot plausibly claim that you are both. However, it is similarly implausible to suggest that you are either one but not the other. There is no significant difference between them, so what could make you one rather than the other?
The best description is to say that you are neither surviving person. This may seem odd. You would have survived if just one transplant had been successful. But this result was even better - not one success, but two! Parfit explains (p.262):
You will lose your identity. But there are at least two ways of doing this. Dying is one, dividing is another. To regard these as the same is to confuse two with zero. Double survival is not the same as ordinary survival. But this does not make it death. It is further away from death than ordinary survival.
This entire scenario seems unexplainable for non-reductionists. Treating identity as a 'further fact' makes it impossible to apply it sensibly to the case of division. But for reductionists it is much easier. We can recognise that we know all the facts (e.g. there will be two resulting people, both psychologically continuous with me), and we can choose how best to describe it in terms of 'identity' talk. The best way to describe it is to say that we are identical to neither resulting person. But there is no rational significance to what we call things. Since the application of 'identity' talk becomes fairly arbitrary in such cases, it shows that identity is not what matters. Rather, it is the underlying facts involving psychological continuity (etc.) that matter. If we divide, then we lose our identity. But this doesn't matter, because people survive who stand in the right psychological relation to us. It cannot be called 'identity' because the relation does not hold uniquely. But the thought experiment shows that it is this relation, and not unique identity, that really matters.