Reductio ad absurdum (Latin: "reduction to the absurd") is a form of argument in which a proposition is disproven by following its implications logically to an absurd consequence.
A common species of reductio ad absurdum is proof by contradiction (also called indirect proof) where a proposition is proven true by proving that it is impossible for it to be false. For example, if A is false, then B is also false; but B is true, therefore A cannot be false and therefore A is true. This of course only works if it is not a false dichotomy; i.e., if there are other choices than true and false (such as undefined, or something entirely different), then this form of proof is a logical fallacy.
Legal and everyday use
Consider the following statement, attributed to physicist Niels Bohr: "The opposite of every great idea is another great idea." If this statement is true, then it would certainly qualify as a great idea - it would automatically lead to a corresponding great idea for every great idea already in existence. But if the statement itself is a great idea, its opposite ("It is not true that the opposite of every great idea is another great idea") must also be a great idea. The original statement is disproven because it leads to an absurd conclusion: that an idea can be great regardless of whether it is true or false.
Some legal usage, and some common usage, depends on a much wider definition of reductio ad absurdum than proof by contradiction, where it is argued a proposition should be rejected because it has merely undesirable (though perhaps not actually self-contradictory) consequences. In a strict logical sense, this might be reductio ad incommodum rather than ad absurdum - since in formal logic, 'absurdity' applies only to impossible self-contradiction.
For example, consider the proposition Cuius est solum eius est usque ad coelum et ad inferos (literally: 'for whoever owns the soil, it is theirs up to Heaven and down to Hell'). This is also known as ad coelum. A legal reductio ad absurdum argument against the proposition might be:
Suppose we take this proposition to a logical extreme. This would grant a land owner rights to everything in a cone from the center of the earth to an infinite distance out into space, and whatever was inside that cone, including stars and planets. It is absurd that someone who purchases land on earth should own other planets, therefore this proposition is wrong.
(This is a straw man fallacy if it is used to prove that the practical legal use of "ad coelum" is wrong, since ad coelum is only actually ever used to delineate rights in cases of tree branches that grow over boundary fences, mining rights, etc. Reductio ad absurdum applied to ad coelum is, in this case, claiming that ad coelum is saying something that it is not. The reductio ad absurdum above argues only against taking ad coelum to its fullest extent.)
It is only in everyday usage that this could acceptably be called a reductio ad absurdum: it is simply reductio ad absurdum being applied to an originally flawed reductio ad absurdum argument where the extremes were not rational for the original proposition.
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