A method of proof that assumes the negation of the target statement and derives a logical contradiction; from the contradiction one concludes the original statement (classically using the law of excluded middle or double-negation elimination).

Principle

Exploit classical logic principles: if ¬P implies a contradiction, then ¬¬P holds, and in classical logic this yields P. The tactic transforms impossibility of the negation into truth of the assertion.

Demonstration

Proving the irrationality of √2: assume √2 is rational, derive an arithmetic contradiction about evenness of numerator and denominator. Another example: show there is no smallest positive rational with a given property by assuming existence and deriving impossibility.

Misapplication

Using contradiction to claim constructive existence without providing a witness (in constructive settings ¬¬∃x is weaker than ∃x), or assuming that a derived contradiction in an informal argument always identifies the precise faulty assumption rather than only ruling out the negation.

Consequence

Allows proofs of negations, impossibility statements, and many existence results in classical mathematics; widely useful when a direct constructive approach is hard or unknown.

Reversal

Contrast with direct or constructive proofs that build the object asserted to exist; contraposition is a related but distinct technique (prove ¬Q⇒¬P to deduce P⇒Q) and is not identical to reductio arguments.

Boundary

Valid in classical logic; in intuitionistic or constructive logics one may only conclude ¬¬P from ¬P⇒⊥, not P itself unless additional principles are accepted. The method applies only when a genuine contradiction (derivation of both A and ¬A or an absurdity) is obtained.

Semantic Tension

Proof by contradiction versus constructive proof: both aim to establish truth but differ on delivering witnesses and on reliance on excluded middle; versus contraposition: contraposition transforms an implication, while reductio assumes negation and seeks absurdity.

Synthesis

Proof by contradiction turns the impossibility of the negation into a proof of the claim by deriving an inconsistency under classical logic; it is a powerful, sometimes nonconstructive, technique complementary to direct and constructive methods.