@Didymos Thomas,
The existence and continuity of the self, and its relationship to moral responsibility are profoundly interesting questions.
Descartes was first to raise this question. He did it as part of his programme to put all knowledge on a basis of absolute certainty by systematically doubting everything it is possible to doubt so that he could be certain at least of what remained. In the end, the only thing he found it impossible to doubt was his own existence, with his conclusion "I think therfore I am." So Descartes' position is that you must exist in order to ask the question. This does seem initially convincing but it has been objected that Descartes has concluded too much: when he thinks, all he is logically entitled to conclude is that there is thinking going on. To attribute it to a self is to assume what he started out to prove.
The same criticism was taken up by Hume. It is worth quoting his conclusion:
"When I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold or light or shade, love or hatred, pain or pleasure. I can never catch myself at any time without a perception. When my perceptions are removed for any time, as by sound sleep, so long am I insensible of myself, and may truly be said not to exist."
Anyone can repeat Hume's procedure. If you try to introspect you will find that you are always aware of some particular thought, feeling or image. It is impossible to be conscious of the actual observing self.
Hume therefore concluded that the self is not a single entity but a bundle of thoughts, feelings and perceptions. This conclusion is similar to the Buddhist position that the individual person is a combination of several distinct factors. Buddhist writing illustrates this by an analogy with a cart, which is composed of several constituent parts. The cart is simply the sum of its constituent parts.
It does not follow from this that the self or the cart are any kind of illusion. The parts of the cart do exist. The whole cart is simply these parts viewed in a certain way. The only mistake would be to assume that there is something else which is added (eg "spiritually") to the components of the cart or the self in order to make the whole.
Leaving Hume and Buddhism behind for my own thoughts, it seems to me that there
is something else added to the components of the cart to enable us to regard it as a whole single thing rather than just its components. What is added is our background awareness of the function of the cart. Our knowledge of what the cart is
for enables us to overlook its constituents and regard it as a single entity.
Applying this approach to the self, I should then ask: What is the function of the self? There is no single answer to that question but it appears to me that an important factor in the emergence of our sense of self is precisely our developing awareness of personal responsibility. Moral rules, at least at the foundation stage of our acquisition of a sense of morality, are attributed to us by others - initially by parents or comparable adults and later by the peer group. The continuing self can be regarded as the social assumption of continuing responsibility for our actions which we initially learn from others and then identify with. By this reasoning, the self is (to a large extent) moral responsibility which is attributed to us by others which we subsequently adopt.
Peter