Proof by Contradiction

Theorem
The proof by contradiction is a theorem of natural deduction.

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

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

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

Tableau Form
In a tableau proof, the proof by contradiction can be invoked in the following manner:
 * Abbreviation: PBC
 * Deduced from: The pooled assumptions of $\bot$.
 * Discharged assumption: The assumption of $p$.
 * Depends on: The series of lines from where the assumption of $p$ was made to where $\bot$ was deduced.

Explanation
If we know that by making an assumption $p$ we can deduce a contradiction, then it must be the case that $p$ cannot be true.

Thus it provides a means of introducing a negation into a sequent.

Also known as
This is also known as not-introduction, and can be seen abbreviated as $\neg \mathcal I$ or $\neg_i$.

However, there are technical reasons why this form of abbreviation are suboptimal on this website, and PBC (if abbreviation is needed at all) is to be preferred.

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

Also see

 * Reductio Ad Absurdum, otherwise known as indirect proof, which has the form $\left({\neg p \vdash \bot}\right) \vdash p$.