Proof by Contradiction

Proof Rule
The proof by contradiction is a valid deduction sequent in propositional logic: If, by making an assumption $p$, we can infer a contradiction as a consequence, then we may infer $\neg p$.

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

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

Explanation
If we know that by making an assumption $p$ we can deduce a contradiction, then it must be the case that $p$ cannot be true.

Thus it provides a means of introducing a negation into a sequent.

Also known as
This is also known as not-introduction, and can be seen abbreviated as $\neg \mathcal I$ or $\neg_i$.

However, there are technical reasons why this form of abbreviation are suboptimal on this website, and PBC (if abbreviation is needed at all) is to be preferred.

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

Also see

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

Technical Note
When invoking Proof by Contradiction in a tableau proof, use the Contradiction template:



where:
 * is the number of the line on the tableau proof where the Proof by Contradiction is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the start of the block of the tableau proof upon which this line directly depends
 * is the end of the block of the tableau proof upon which this line directly depends