Another thought experiment to put the problem into focus, originated by Derek Parfit and with modifications from Joe Kern:
We know that it's possible to live with only half a brain. In principle, there is nothing preventing someone with only a half a functioning brain from getting a transplant to fill the other half. They would feel different after the operation, but if done properly their memories would remain intact, so they would experience
undergoing a change (perhaps in personality, mood, temperament) rather than ceasing to exist. Presumably, with the right technology, the same thing could be said of a partial or entire brain transplant; for the sake of argument, let's assume that if your brain was successfully transferred to the body of someone physically identical to you, it would feel like "waking up" in that body.
Joe Kern said:
(Case 1) If one hemisphere of your brain dies, the right lets say, then you will continue to live and have a conscious existence based just on the functioning of the left hemisphere. If this left hemisphere is then transplanted into a new body (a second body) identical to your original body, you will continue to exist from that left hemisphere in that new body, just as if your whole brain had been transplanted into a new body.
But, what if:
(Case 2) Your whole brain is functioning, and it is removed from your body, but before it is transplanted into the new body, the two hemispheres are split from each other, and the right hemisphere is destroyed? The result is the same as in (Case 1): just your left hemisphere is in a new body. It should not matter whether the right hemisphere died while still attached to your left hemisphere in your old body, or after it was detached outside your body. You should therefore believe as you did in (Case 1), that you will exist in that new body, just off the functioning of your left hemisphere.
So now, what if:
(Case 3) Your whole brain is functioning, and it is removed from your body, and the two hemispheres are split from each other (the same as the previous case so far), but instead of the right hemisphere being destroyed, it is transplanted into another (a third) body identical to your original body? From the point of view of your left hemisphere, what happens to the right hemisphere after splitting should not matter. Once the two are separated, to Lefty (which is the person who is the left hemisphere in the new body) the fate of the right hemisphere is an extrinsic fact. So, if you become Lefty in (Case 2), then you should also become Lefty in (Case 3).
But here we have the problem, because this exact same story could be told from the perspective of Righty, and we would conclude that you should become Righty in (Case 3). There are competing claims for who you will become in (Case 3).
Does the problem make sense? The way I look at it, there is a fact of the matter, from my perspective, about which person I will "wake up" as. Let's suppose one body is in Syria and the other is in Denmark, and afterwards they have to live the remainder of their lives in these countries. Before the operation, I would just be in one place. After the operation, I would still be in just one place: either Syria or Denmark. I wouldn't experience being two people at once, in two different places at once. Rationally, I would hope to wake up in Denmark rather than Syria, as these are tangibly different outcomes for my well-being. There doesn't seem to be any way, even in principle, to physically predict where I would end up, so I ought to conclude that it's not a real question. However, prior to the split, it seems incontrovertibly, actually true that I should either look forward to waking up in Denmark or dread waking up in Syria, since I can't possibly experience being in both places at the same time.
It's this me-ness that I'm trying to get at, isolated from the extrinsic facts about my body and brain. On the one hand, there shouldn't be such a thing if dualism is false, and I think it is. On the other hand, there is no answer to the above riddle that doesn't involve something distinct from the physical world. How to resolve this paradox?