Reductio ad Absurdum

Proof Rule
The reductio ad absurdum is a valid deduction sequent in propositional logic: 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:
 * $\displaystyle {\begin{array}{|c|} \hline \neg p \\ \vdots \\ \bot \\ \hline \end{array} \over p} \textrm{RAA}$

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.

Technical Note
When invoking Reductio Ad Absurdum in a tableau proof, use the Reductio template:



where:
 * is the number of the line on the tableau proof where the Reductio Ad Absurdum is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the start of the block of the tableau proof upon which this line directly depends
 * is the end of the block of the tableau proof upon which this line directly depends