Reductio ad absurdum (from Latin Reduced to an absurdity) is a tool of logic, also known as "proof by contradiction".
Say we wish to prove hypothesis A. The procedure is to show that assuming "not A" (i.e. that A is false) leads to a logical contradiction. Thus A cannot be false, and must therefore be true. For examples, see Irrational number and Cantor's diagonal argument.
- basically: if
- S union { ¬ t } |-- F
- then
- S |-- t
- S union { ¬ t } |-- F
It is important to note that to form a valid proof, it must be demonstrated that "not A" implies a property that is actually false or actually incompatible with "not A". The danger here is the logical fallacy of argument from lack of imagination, where it is proven that "not A" implies a property "B", which looks false, but is not really proven to be false. Historical examples of this fallacy include false proofs of the fifth postulate a.k.a. parallel postulate of Euclid (see Non-Euclidean geometry).
See Symbolic logic.