reductio ad absurdum

In formal logic, reductio ad absurdum (Latin: "reduction to the absurd") is an argument to refute a proposition (or set of propositions), by showing that it leads to a logically absurd consequence.^[1] That is, the proposition is shown by proper inspection to be simply untenable within the rules of logic, because it necessarily leads to a self-contradictory consequence.

Consider the proposition X is an even prime number greater than 2 that follows the basic laws of mathematics. From that can be derived that:

1. X is not 2 (since it was greater than 2)
2. X is not divisible by any other number than 1 or itself (because it was a prime number)
3. X is not divisible by 2 (follows from 1. and 2.: 2 is not X or 1)
4. but X is divisible by 2 (because X is even)
5. thus X both is and is not divisible by 2

This conclusion is obviously self-contradictory - that is, logically absurd (it violates the law of non-contradiction) - requiring the rejection of the original proposition as false.

Legal and everyday use

Some legal usage, and some common usage, depends on a much wider definition of reductio ad absurdum, 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. ^[1]

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.^[2] Reductio ad absurdum applied to Ad Coelum is, in this case, claiming that Ad Coelum is saying something that it is not. If anything, the reductio ad absurdum above argues only against taking Ad Coelum to its fullest extent.)

It is only in everyday (or legal) usage that this could acceptably be called a reductio ad absurdum.

See also

• Proof by contradiction
• Devil's advocate

References

1. ^ ^{a b} Nicholas Rescher. "Reductio ad absurdum". The Internet Journal of Philosophy. http://www.iep.utm.edu/r/reductio.htm. Retrieved 21 July 09. 
2. ^ "Definition of Land Chapt. 2". OUP UK. http://www.oup.com/uk/orc/bin/qanda/sample_chapters/wilkie_chap02.pdf.  "The owner’s rights extend to such a height as is reasonably necessary for the ordinary use and enjoyment of the land. In Baron Bernstein of Leigh v Skyviews and General Ltd [1978] QB 479, Griffith J stated that it was necessary to balance the rights of an owner to enjoy the land against the rights of the general public to take advantage of all that 'science now offers in the use of airspace'."

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