wittgenstein said:
please explain how contradicting
Law of excluded middle - Wikipedia is not extraordinary.
Classical logic is not the only possible logic. It has many undesirable properties, such as being explosive (one contradiction destroys the whole system). For a general overview of other possibilities, see Graham Priest,
An Introduction to Non-Classical Logic, 2nd Ed.
Consider a monotonic database of facts (i.e. you can add new facts, but never delete old ones) that starts empty. You also have a set of axioms and allowable proof methods. You are allowed to add facts if either (1) you learn them externally (from some outside source) or (2) you can prove them using the existing facts, axioms, and methods. A fact is TRUE if it is in the database. A fact is FALSE if its negation is in the database. It is quite clear that these are not the only two possibilities. A fact is NEITHER if it is neither TRUE nor FALSE; in an empty database, all facts are NEITHER. A fact can also be BOTH, i.e. contradictory. So there are 4 logic values, and the NEITHER value violates the law of the excluded middle. It
is the excluded middle. Yet this is a perfectly sensible logic system.
Note that any logic system which is paraconsistent (doesn't explode on 1 contradiction) cannot be an extension of classical logic and contain it as a subset, because the subset would still explode. It instead must give up at least one part of classical logic. In the above, we have given up both Excluded Middle and Non-contradiction (including proof by contradiction).
Another example:
Discursive logic models a room full of people with different beliefs. A statement is TRUE if at least one person consistently believes it. So "Trump was a good president" and "Trump was a bad president" can both be TRUE if there are different people who consistently believe one or the other. However "Trump was a good president & Trump was a bad president" cannot be consistently believed by anyone. Thus in this system we have given up conjunction: "A" being true and "B" being true does NOT imply that "A & B" is true. So you can have "A" and "~A" both be TRUE, and still not be able to construct the contradiction "A & ~A" and explode.
At any rate, the Law of the Excluded Middle is missing from many paraconsistent logics, and can not at all be assumed to be universally true or applicable. In particular, there is no reason to assume that it
must apply to quantum logics, which we already know have many non-intuitive properties (like negative probabilities).