Every person is one person. Everyone is self-identical.
The name, "Sam Clemens" may refer to a number of different people, but how can that possibly mean (whatever that might mean) that Sam Clemens is a number of different people. The name has nothing to do with what is named. If it makes if clearer for you, let's instead of the name "Sam Clemens" simple have, "Sam Clemens-1", "Sam Clemens-2", "Sam Clemens-3", ...."Sam Clemens-n". Each of those "Sam Clemens" is exactly one person. The only problem is that you are mixing up the name with what it names. There are many names, but each of the names is the name of just one person.
For the sake of logic, an identity only has to be tenative. Once it is established, firmly and without doubt, then it is an axium that it does not change; that A is A, and B is B.
I don't know if I am logical. I do know I am mechanical. I have spent a great deal of time learning a lot of stuff through reading. When some one, anyone, tries to make a statement, a judgement concerning reality, as a truth, or truism, then if there is an exception when compared to what I know, I can point it out. I love aphorisms, and not because they are always true, but often true, and not always illuminating, but often illuminating.
If we say logic has a purpose, and serves that purpose it does not mean it is infallable. The syllogism is seriously weak, yet still effective at what it does, which is to give serious reasoning something to work on -that is, more than a total guess. Insight is essential to discovery, and perhaps an essential part of human intelligence. What is it without scientific method, deduction, and induction? The Syllogism is an informed jump to conclusions. An insight is an informal jump to conclusion. Once there, the fault finding, negative power of logic will help to tell if you are on firm ground.
Most logic seems totally useless to creativity, discovery, or insight. Rather; reason requires a set table before it can break dishes. Kant was very logical, in my opinion, and insightful; yet people of his age called him a great destroyer because of what he did with his logic to metaphysics. But that could have happened anytime in the previous five hundred years if anyone had turned logic loose on what they took first, and foremost, for granted.