Reductio ad absurdum

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical argument, or proof by contradiction, is the form of argument that attempts to establish a claim by showing that following the logic of a contrary proposition or argument would lead to absurdity or contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof.

This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called proof by contradiction. In formal logic, this technique is captured by an inference rule for reductio ad absurdum.

More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile.

Examples

The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as can be seen in the following examples of refutation by contradiction:

The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find people falling off the edge.

There is no smallest positive rational number; if ⁠

q

{\displaystyle q}

⁠ were the smallest positive rational, then ⁠

q

/

2

{\displaystyle q/2}

⁠ would be a positive rational number that is smaller than ⁠

q

{\displaystyle q}

⁠ because it equals half of ⁠

q

{\displaystyle q}

⁠, and it would also not be smaller than ⁠

q

{\displaystyle q}

⁠ because ⁠

q

{\displaystyle q}

⁠ is assumed to be the smallest positive rational.

The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses (empirical evidence).