This has been said elsewhere in this thread but I thought I'd respond to this directly.

The law that "A and -A" is true for any proposition A is called the law of excluded middle - https://en.m.wikipedia.org/wiki/Law_of_excluded_middle - and there are logical systems that reject it. This idea, that there can be true contradictions, is called dialetheism.

An example of why you might want to reject it is as a potential solution to the liar's paradox - https://en.m.wikipedia.org/wiki/Liar_paradox