##  [Proof by Contradiction](/proof-contradiction-0) 

 Definition

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

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

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

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

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

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

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

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

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.