Although Elizier has dealt with personal identity questions (in terms of ruling out the body theory), he has not actually, as far as I know, "solved" the problem of Personal Identity as it is understood in philosophy. Nor, as far as I know, has any thinker (Robin Hanson, Yvain, etc) broadly in the same school of thought.
Why do I think it worth solving? One- Lesswrong has a tradition of trying to solve all of philosophy through thinking better than philosophers do. Even when I don't agree with it, the result is often enlightening. Two- What counts as 'same person' could easily have significant implications for large numbers of ethical dilemnas, and thus for Lesswrongian ethics.
Three- most importantly of all, the correct theory has practical implications for cryonics. I don't know enough to assert any theory as actually true, but if, say, Identity as Continuity of Form rather than of Matter were the true theory it would mean that preserving only the mental data would not be enough. What kind of preservation is necessary also varies somewhat- the difference in requirement based on a Continuity of Consciousness v.s a Continuity of Psyche theory, for example should be obvious.
I'm curious what people here think. What is the correct answer? No-self theory? Psyche theory? Derek Parfit's theory in some manner? Or if there is a correct way to dissolve the question, what is that correct way?
I think it helps to look at statements of personal narrative and whether they're meaningful and hence whether they can be true or false. So, for example, change is part of our personal narrative; we mature, we grow old, we suffer injuries, undergo illness, etc. Any philosophical conception of personal identity that leads to conclusions that make change problematic should be taken as a reductio ad absurdum of that conception and not a demonstration of the falsity of our common sense concepts (that is, it shows that the philosopher went wrong in attempting to explicate personal identity, not that we are wrong). Statements of personal narrative are inclusive of our conception, birth, events of our life, etc. Most cultures give meaning to post-death statements but it's a clearly differentiated meaning. But I can't meaningfully speak of being in two places at once, of being destroyed and recreated, of not existing for periods of time, etc, so a large range of philosophical and science fiction scenarios are ruled out. (Again, if a philosopher's attempt to explicate personal identity makes these things possible then it is the philosopher who erred, since the common sense concept clearly precludes them; or he/she is now using a novel concept and hence no pertinent inferences follow). If we create a new person and give him the memories of a dead man, we have only played a cruel trick on him, for a statement of personal narrative that includes being destroyed and recreated has no sense ("I didn't exist between 1992 and 1998" isn't like "I was unconscious/asleep between 1992 and 1998" because non-existence is not a state one can occupy).
Note that the meaningfulness of novel statements like "I teleported from Earth to Mars" or "I uploaded to Konishi Polis in 2975" depend entirely on unpacking the meaning of the novel terms. Are "teleported" and "uploaded" more like "travelled" or more like "destroyed and recreated"? Is the Konishi Polis computer a place that I can go to? The relevant issue here isn't personal identity but the nature of the novel term which determines whether these statements are meaningful. If you start from the assumption that "I teleported from Earth to Mars" has a clear meaning, you are obviously going to come to a conclusion where it has a clear meaning. Whether "teleported" means "travelled" or "destroyed and recreated" does not turn on the nature of personal identity but on the relationship of teleportation to space - i.e., whether it's a form of movement through space (and hence travel). If it involves "conversion from matter to information" we have to ask what this odd use of "conversion" means and whether it is a species of change or more like making a description of an object and then destroying it. The same is true of uploading. With cryonics the pertinent issue is whether it will involve so much damage that it will require that you are recreated rather than merely recovered.