Reductio ad Absurdum/Variant 2/Proof 1

Theorem

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

Proof
By the tableau method of natural deduction:

Comment
It can be seen that this result depends on the rule of Double Negation Elimination.

As this depends on the Law of the Excluded Middle, it invalidates the Reductio Ad Absurdum from the intuitionist system.

Also see
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.

Linguistic Note
Reductio ad absurdum is Latin for reduction to absurdity.