Reductio ad Absurdum

Proof Rule
The reductio ad absurdum is a theorem of natural deduction.

If, by making an assumption $\neg p$, we can infer a contradiction as a consequence, then we may infer $p$.

The conclusion does not depend upon the assumption $\neg p$.

It can be written:
 * $\displaystyle {\begin{array}{|c|} \hline \neg p \\ \vdots \\ \bot \\ \hline \end{array} \over p} \textrm{RAA}$

Tableau Form
In a tableau proof, the reductio ad absurdum can be invoked in the following manner:
 * Abbreviation: $\textrm{RAA}$
 * Deduced from: The pooled assumptions of $\bot$.
 * Discharged assumption: The assumption of $\neg p$.
 * Depends on: The series of lines from where the assumption of $\neg p$ was made to where $\bot$ was deduced.

Explanation
If we know that by making an assumption of the falsehood of $p$ we can deduce a contradiction, then it must be the case that $p$ must be true.

Variants
The following forms can be used as variants of this theorem:

Also see

 * Clavius's Law


 * Proof by Contradiction, often treated as another aspect of the same thing.

From the point of view of purely classical logic, this is acceptable. However, in the context of intuitionistic logic, it is essential to bear in mind that only the Proof by Contradiction is valid.

Linguistic Note
Reductio ad absurdum is Latin for reduction to absurdity.