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.
Eschew all obfuscation!
| 
