**Proof by contradiction**, or *reductio ad absurdum* is an argument against a proposition or formula in logic which shows that the formula in question results in a contradiction.

In a proof by contradiction, a negation is obtained by proving that its opposite formula creates a contradiction. The method of proof by contradiction is used for both the **negation introduction** (¬I) and **negation elimination** rules of inference, and explains that if "P" results in some contradiction, then ¬P must be true.

Proof by contradiction may be formally presented as follows:

**(α → ⊥) ¬α**

In this example, the symbol ⊥ is used to indicate a contradiction. Alternatively, we may formulate this by showing an actual contradiction of two formulas:

**(α → (β ∧ ¬β)) ¬α**

These examples show the negation introduction (¬I) version of the proof by contradiction, which introduces a negation. The negation elimination (¬E) version is the inverse:

**(¬α ⊥) α**

When beginning with a negation, we have the choice of choosing to conclude either α (by ¬E) or ¬¬α by (¬I), since α and ¬¬α are equivalent, according to the double negation rule.