Proof by Contradiction

Context
This 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 $$\lnot p$$:

$$\left({p \vdash \bot}\right) \vdash \lnot p$$

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

This is sometimes known as "`not-introduction".


 * Abbreviation: $$\lnot \mathcal{I}$$
 * 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 $$\left({\lnot p \vdash \bot}\right) \vdash p$$. 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.