The important thing to note is that it doesn't have anything to do with truth or truth-values of propositions. It breaks the fundamental operation of the prova

The important thing to note is that it doesn't have anything to do with truth or truth-values of propositions. It breaks the fundamental operation of the provability of a statement. And, since many proofs are done by assuming a statement's inverse and trying to prove a contradiction, having a known contradiction in the set of provable statements can effectively allow any statement to be proven. Keeping the contradiction is not actually an option.