Q: Is there a logical proof that all souls are multiple personalities of the same self, and of what therefore to do?
A: What an interesting question. I am not a logician, but I will try giving an answer that might intrigue you.
All there is, is truth. What is, is what is! However, not all truths are equal as some truths are small and some are big. A purely conceptual thought, like say a mythical creature with seven heads and purple polka-dots all over its furry body is a very small truth. It only exists in my head (and maybe in yours now too). Physical objects have more truth or reality, and in my view, the soul has even more reality. The sense that we are all one is an even bigger truth.
I define the truth as whatever opens your heart or sense of being and quiets the mind. A smaller truth has the opposite effect. That is not exactly a logical definition, but it is an effective way to sort the many truths we encounter in life by how true they are, or in other words, how big a truth they are.
So back to your question. Let's propose three different truths and see how they measure up:
1- We are separate individuals and when we die, that is that.
2- We are individuals with souls that continue on after we die and then reincarnate in a new body.
3- We are all one soul that incarnates in all of the bodies throughout time.
Which one opens or relaxes your awareness the most fully? Which one quiets your mind the most? Just try them on like you are trying on a piece of clothing and see which one fits the most spaciously. That is the truest conclusion. And by the way, this is always contextual and includes what fits with all of your other beliefs and experiences, so what is true for you at this point in time may be different than what is true for someone else, or for you at another point in time.
There is a fourth possibility:
4-All three truths above are true simultaneously at different levels of experience or reality.
So, depending on your perspective and where you are looking from, you can experience yourself as a completely separate being with little or no fundamental connection to other people. Or with a shift in perspective or where you are experiencing your self from, you may then sense that you as an individual continue through space and time beyond the time of your birth and death. Then with a very big shift in perspective, you may experience that you are one with all souls and all expressions of life as a whole. However if all of these perspectives exist simultaneously, then you will also be able to shift back into a smaller perspective and experience your self as a completely separate and seemingly purely physical entity. For example, that is likely to happen if you have a tooth drilled at the dentist with no novacaine!
There is even a level of truth or reality where all of the above is seen to be a dream and completely unreal. At that level, there are no bodies, no souls, no one infinite soul and nothing ever happens. Even this empty level of truth can coexist with the other levels, so it still might be a matter of your perspective in the moment.
There are lots of wonderful metaphors for this idea of simultaneous different levels of truth, including sometimes contradictory truth. One is how on a rainy day, you can look up and see a sky filled with grey clouds. And yet you can fly in airplane over that same spot and experience blue sky and sunshine. And you can even go up in a rocket ship and discover a black sky and a sun that never sets. And you can even continue out into the empty blackness of outer space beyond the distance at which you can perceive either the earth or the sun or even our entire galaxy! And yet all of these levels of the truth exist simultaneously, even if they might seem logically to contradict each other.
And then there can be lots of smaller truths within any bigger truth. Someone a few hundred miles away from you on the surface of the earth might be experiencing different weather, or even nighttime with no sun. Same level, but still very different truths.
In physics, it would seem that physicists are struggling with a similar dilemma as they try to sort out the different rules that apply to ordinary levels of experience versus a much larger scale where Einstein's theory of special relativity explains some very strange and different truths. Or they can explore the extremely small dimensions where quantum theory describes an even stranger set of rules for how things happen in this reality. Again, these different levels all coexist at the same time, even though there are radical contradictions between the different rules or theories.
How many levels you distinguish and where you draw the line between different levels is somewhat arbitrary, but I would invite you to also check out Timothy Conway's excellent model that proposes three different levels of nondual reality here: http://www.enlightened-spirituality.org/3_levels_of_nondual_Reality.html
As for what to do, that of course depends on what level of truth you are experiencing in the moment. Thankfully there are lots of different answers to that question also. If there were not different truths, and even different sizes or levels of truth, then it seems that this life would be unbearably homogenous and there would be little opportunity for endless debates and discussions on the internet! As it is, we can seemingly never be done exploring the truth from all different perspectives and levels, which may be just what consciousness as a whole is up to when it creates all these apparent individuals.
I hope this is helpful, although I would guess it all raises more questions than it answers :)