Allow one contradiction into classical logic and you can prove absolutely anything
Logicians call it the principle of explosion: once a set of axioms contains a contradiction, every proposition becomes provable. Latin gives it a cheeky name, ex falso quodlibet, meaning that from falsity anything follows. The same idea powers one of mathematics' favourite weapons, proof by contradiction.
In traditional logic a contradiction is a proposition that clashes with itself or with established fact, and spotting one has long served as a way to expose bias or bad faith. The law of noncontradiction holds that a thing cannot both have and lack the same property, at the same moment and in the same respect. Modern formal logic and type theory narrow the word: a proposition counts as contradictory if false can be derived from it, making it false under every condition.
Plato saw the need for the concept and dramatised it in his dialogue Euthydemus. There the sophist Dionysodorus insists that false opinion and ignorance do not exist, and dares Socrates to refute him. Socrates points out the trap: if nobody can ever say anything false, refutation itself becomes impossible, so the thesis undermines the very debate it is offered in.
Mathematicians turn this into a tool. To show a statement true, assume its negation and derive an absurdity; the assumption must then be wrong. The irrationality of the square root of 2 is the classic case, awkward to establish head-on but quick once you suppose it rational and watch the supposition collapse. Proofs of this kind often end with a special contradiction symbol followed by Q.E.D. The method depends on the law of excluded middle, and in some settings it is inefficient or unavailable.
Logicians also study weaker systems. Minimal logic keeps most classical axioms but drops explosion and proof by contradiction; adding explosion back yields intuitionistic logic, while adding double-negation elimination restores full classical logic. Contradiction even troubles consistency proofs. Emil Post, extending his 1921 consistency proof for propositional calculus beyond Principia Mathematica, realised that general postulate systems might not contain negation at all, so he needed a fresh definition of consistency.
Source: Contradiction
