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.