I think your problem here is that you are confusing common language (english) with more formal language, ie logic and mathematics.
In logic and mathematics you can provide proofs with reasoning. Deductive, Inductive etc...
But this has nothing to do with the common usage of the word 'impossible', which as far as I know has no formal definition. That is, the fact people say x is impossible is no different from them saying they can't believe/imagine x is true. You can't scientifically call something impossible, only observed or not observed, probable or improbable.
As to the Rand nonsense, identity is a matter of definition, so x is only x to the degree that you have a definition of x. If you don't define your premises, or even fail to define them correctly, then no amount of logic will save you or give you something that is not contradictory. Add to that, the logic you use will be based on your experience and you are left with the same old nasty problem of induction.
Schrodingers cat is a good example of this, since in that situation you actually have 3 choices: x, notx, xandnotx. Now this only appears to be the case on the quantum level, but the point is, we form systems of logic based on our observations, change the context of those observations in a radical fashion and our current logic will likely fail.
"How many angels can dance on the head of a pin?
As many as want to..."