Reductio ad Absurdum

Proof Rule
Reductio ad absurdum is a valid argument in certain types of logic dealing with negation $\neg$ and contradiction $\bot$.

This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.

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

Also see

 * Definition:Logical Not
 * Definition:Contradiction


 * Clavius's Law


 * Proof by Contradiction, often treated as another aspect 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.

This is because Proof by Contradiction starts with a positive assumption $\phi$.

As a result, it does not depend on the Law of Excluded Middle.