Proof by Contradiction

Theorem
The proof by contradiction is a theorem of natural deduction.

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

Sequent Form
The proof by contradiction is symbolised by the sequent:
 * $\left({p \vdash \bot}\right) \vdash \neg p$

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

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

Tableau Form
In a tableau proof, the rule of or-elimination can be invoked in the following manner:


 * 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.

Also known as
This is also known as not-introduction.

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.

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

Also see

 * Reductio Ad Absurdum, otherwise known as indirect proof, which has the form $\left({\neg p \vdash \bot}\right) \vdash p$.