Reductio ad Absurdum

Context
Natural deduction.

Definition
If, by making an assumption $$\lnot p$$, we can infer a contradiction as a consequence, then we may infer $$p$$:

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

Its abbreviation in a tableau proof is $$\textrm{RAA}$$.

It is also referred to as indirect proof.

Proof
First, from the Rule of Implication we see that:

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

Then we show that $$\lnot p \Longrightarrow \bot \vdash p$$:

Thus:

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

Comment
Because of their similarity in form, many authors treat the Reductio Ad Absurdum and the proof by contradiction as two aspects 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.