Proof by Contradiction

Context
The proof by contradiction is one of the axioms of natural deduction.

The rule
If, by making an assumption $$p$$, we can infer a contradiction as a consequence, then we may infer $$\neg p$$:
 * $$\left({p \vdash \bot}\right) \vdash \neg p$$

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

This is also known as not-introduction.

It can alternatively be rendered:


 * $$p \implies \left({q \and \neg q}\right) \vdash \neg p$$

from the definition of contradiction.

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


 * Abbreviation: $$\neg \mathcal{I}$$
 * 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
This means: if we know that by making an assumption $$p$$ we can deduce a contradiction, then it must be the case that $$p$$ can not be true.

Thus it provides a means of introducing a logical not into a sequent.

Comment
Note the similarity between this and Reductio Ad Absurdum, otherwise known as indirect proof, which has the form $$\left({\neg p \vdash \bot}\right) \vdash p$$.

The latter is strictly speaking not axiomatic, as it requires the acceptance of the Law of the Excluded Middle which is not accepted by the school of intuitionist logic.