Proof by Contradiction

A precise guide to contradiction proofs, correct negation, and identifying the impossible conclusion.

Claim → evidence → conclusion. A claim or task is translated into assumptions or inputs and a target or output, supported by a finite chain of valid steps, and closed by reconnecting the result to the original goal.

Precise definition

To prove a statement $P$ by contradiction, assume its negation $\neg P$ and derive a contradiction: a statement known false, such as $R\land\neg R$, or a violation of an established definition or theorem. Because $\neg P$ cannot hold consistently with the accepted premises, $P$ must be true.

Notation and mathematical language

Negation must respect quantifiers: $\neg(\forall x\,P(x))\equiv\exists x\,\neg P(x)$ and $\neg(\exists x\,P(x))\equiv\forall x\,\neg P(x)$. For an implication, $\neg(P\to Q)\equiv P\land\neg Q$, so both a true premise and false conclusion are assumed.

Conceptual picture

A contradiction proof explores what the world would have to look like if the claim failed. The argument is complete only when that hypothetical world conflicts with the theorem's other hypotheses or accepted facts. A surprising or unlikely consequence is not enough.

Fully worked example

Interpretation and application

Contradiction is powerful for impossibility, uniqueness, irrationality, and infinitude results. In applied arguments, 'the model predicts an impossible value' may instead reveal that a modelling assumption fails; it does not automatically prove a real-world claim outside the model.