Q: Is there a logical proof that all souls are multiple personalities of the same self, and of what therefore to do?
A (Dennis): ‘No’ is the simple answer. If there were, scientists would not still be looking for the origin of consciousness in the brain! It is rather that there exists a body of knowledge from those who have realized that this is how it is. ‘Teachers’ draw on this, together with their own experience, to explain things to seekers until such time as they realize the truth for themselves. To one who has been through this process, there is no problem in understanding that this is perfectly acceptable. To one who has not, however, it seems quite unacceptable and not really any different from the ‘faith’ of religions.
Incidentally, the phrasing of your question indicates that you do not appreciate the ‘bottom line’ message of Advaita. There are no individual ‘souls’ or ‘personalities’ and nothing to ‘do’ in reality. There is only the Self – and you are That (already). You just do not realize this. I.e. all that needs to happen is to remove the ignorance that is preventing you from seeing what is already the case.