If you accept a few simple laws of classical logic, and you accept that a statement and its negation are both true, then everything else is simultaneously true and false. Collquially, "from a contradiction everything is true" or "from a contradiction, everything follows". Sometimes this is known as the Principle of Explosion, because if you accept a contradiction, then your logical system blows up in your face, so to speak.

It's not hard to prove any other statement from a contradiction. Suppose you grant me that both P and its negation -P are true (e.g. it is simultaneously true that I am and am not the Pope). Let R be any other statement (e.g. "roses are blue".) Then, the statement E = "either P or R is true" is itself true because P is true. However, P is also false (i.e. -P is true), thus we conclude that R is true because we have eliminated the case that P is true (this is a disjunctive syllogism). We conclude that if I am and am not the Pope, then roses are blue.

Note that we could have taken the negation "roses are not blue" also, and we could have proven that to be true. So given a contradiction, everything is true, and everything is also false. Our logical system becomes trivial and there's nothing interesting left to prove or disprove.

Now, humans being what humans are, sometimes they are troubled by the notion that from a contradiction everything follows. Unfortunately, the only way to believe a contradiction and not conclude that everything follows from it, is to discard one of the logical laws used to prove this. In the proof that I provided, really the only logical law you could reject is disjuntive syllogism. That's a pretty big law to reject. So, for example, if I was certain that everyone is either dead or alive, and I knew that you weren't dead, then I would be unable to conclude that you're alive. Or if today either is Sunday or it's any other day, and I know it's not any other day, then I can't conclude that today it's Sunday. Either way, you have to reject a lot of classical logic if you want to believe in a contradiction, or you have to accept everything as true or false and give up logic altogether.

The logical systems that accept contradictions, sometimes useful in software, are known as paraconsistent logic. For almost every other day use, though, it is much easier to accept the usual logical laws and to reject all contradictions.

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