it always seemed clear to me that we can not conclude lets say B from A, and also conclude B from ~A. Lets assume that i have a set of sentences S that are logically valid. Over that set of sentences i introduce a sentence A and i get a contradiction, so i say that ~A must be true (reductio ad absurdium). So if ~B was always a fact in S, how can i prove that i can not obtain B from both A and ~A? I know that ~A is the negation of A, but how can i prove that there cant exist a single thing that is true in A and also true in the presence of ~A?
Thanks in advance