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 selfcontradictory 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 the following:
This conclusion is obviously selfcontradictory  that is, logically absurd (it violates the law of noncontradiction)  requiring the rejection of the original proposition as false.
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 selfcontradictory) 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 selfcontradiction. ^{[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. 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.
"Reductio ad absurdum" is a logical rebuttal that takes a proposition to its logical extremes and examines the veracity of the conclusions the proposition implies in those extremes.
In the case of the ad coelum example above, 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.
Reductio ad absurdum is a Latin phrase. It can be translated as reduction to the impossible. Generally, it is also known as Proof by contradiction. In Logic and mathematics it is a method of proving something.
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 of proving something works by first stating something is true. Then other things are deduced from that. In the end, there is a contradiction. This contradiction then shows that thing stated first cannot be true.
