Indirect Proof

Theorem
Let $P$ be a proposition whose truth value is to be proved (either true or false).

There are two aspects to this:

Reductio Ad Absurdum
A reductio ad absurdum argument for the truth of $P$ is a valid argument which takes as a premise the negation of $P$, and from it deduces a contradiction:


 * $\neg P \implies \bot \vdash P$

Proof by Contradiction
A proof by contradiction argument for the falsehood of $P$ is a valid argument which takes $P$ as a premise, and from it directly deduces a contradiction:


 * $P \implies \bot \vdash \neg P$

Proof
For proofs, see:


 * Reductio Ad Absurdum
 * Proof by Contradiction.