If your system of logic contains a contradiction, then it can be said that any statement in that system "leads to" a contradiction. Thus "proof by contradiction" can prove any statement in that system.
Is it contradiction in the proof, or rather a contradiction in assumptions?
More on logic that permits all that *Bounded* Turing Machine does: https://arxiv.org/abs/2106.13309
Why exactly it "leads to"? I don't get it.
Why system can't have a contradiciton for one set of inputs, and not have it for others?