Reductio ad absurdum
Reductio ad absurdum is a Latin phrase which means "reduction to the absurd". The phrase describes a kind of indirect proof. It is a proof by contradiction, and is a common form of argument. It shows that a statement is true because its denial leads to a false or absurd result.
The ridiculous or "absurdum" conclusion of a reductio ad absurdum argument can have many forms. For example,
- Rocks have weight, otherwise we would see them floating in the air.
- Society must have laws, otherwise there would be chaos.
- There is no smallest positive rational number, because if there were, it could be divided by two to get a smaller one.
The phrase can be traced back to the Greek η εις άτοπον απαγωγή (hê eis átopon apagogê). This phrase means "reduction to the impossible". It was often used by Aristotle. The method is used a number of times in Euclid's Elements.
Reduction ad absurdum can be a tool of discovery.
The method of proving something works by first assuming something about it. Then other things are deduced from that. If there is a contradiction, it shows that the first something cannot be correct. For example,
- Weston, Anthony. (2009). A Rulebook for Arguments, pp. 43-44.
- Nicholas Rescher. "Reductio ad absurdum". The Internet Encyclopedia of Philosophy. Retrieved 21 July 09. Check date values in:
- Heath, Thomas Little 1908. The Thirteen Books of Euclid's Elements, Vol. 1, p. 136.
- Polya, Goerge. (2008). How to Solve It: A New Aspect of Mathematical Method, p. 169.