Definition
A contradiction is a pair of statements (or a single compound statement) that cannot both be true at the same time and in the same sense — most directly, a proposition asserted together with its own negation. The law of non-contradiction, which Aristotle treated as the most secure of all principles, holds that no statement can be both true and false simultaneously in the same respect. To assert a contradiction is to assert nothing coherent.
In propositional logic, the contradiction is the statement form P and not-P, which is false on every possible assignment of truth values. Its mirror image is the tautology, which is true on every assignment. Together they mark the two extremes of logical content.
Why it matters
How it works
The destructive power of contradiction comes from a result called the principle of explosion: from a contradiction, any proposition whatsoever can be validly derived. If both P and not-P are accepted, then anything follows — the logical system collapses into triviality. This is why consistency is treated as the minimum demand on any acceptable theory; a theory that contradicts itself loses the ability to discriminate true claims from false ones.
The same principle can be turned into a proof technique. To establish a claim, one can assume its negation and show that the assumption, combined with already-accepted truths, leads to a contradiction. The contradiction is then taken as evidence that the assumption is false, which establishes the original claim. This is reductio ad absurdum, one of the oldest and most reliable strategies in mathematics and philosophy. Euclid used it to prove the infinitude of primes; philosophers use it to undermine positions whose consequences cannot be lived with.