The argumentation from the absurd is a form of indirect evidence based on contradiction. This label includes a family of arguments concluding that an assertion or a proposal should be rejected on the basis of the indefensible consequences which would result from its adoption.

The general operation of reduction to the absurd corresponds to the following mechanism:

1. A claim, a proposal is put forward, as a working hypothesis, a possibility…
2. Consequences are drawn from this proposition, whatever they may be, causal, logical…
3. One of these consequences is deemed to be “absurd” in relation to some criteria, cf. infra
4. The initial proposal or hypothesis is rejected.

Argumentation to the absurd is not an argument from ignorance. An argument from ignorance concludes that P is true because we have failed to prove not-P, whilst an argument to the absurd concludes that P is true because it has been shown that the proposition not-P is false, and that between P and not-P, only one can hold true. This corresponds to a case-by-case argument in a situation where the number of cases is reduced to two: P is true or not-P is true; but not-P is false, so P is true. S. Apagogic argument; Contradiction.

2. Varieties of absurdities

There are as many kinds of reduction to absurdity as modes of deduction and reasons to evaluate a consequence as inadmissible. The qualification as absurd may thus apply to:
— Mathematical consequences. One clearly sees the variety and the diversity of what is called the “absurd” in argumentation by contrasting these forms with the demonstration from the absurd, where absurd means “contradictory”, cf. infra.

— Logical or semantical consequences. The consequences analytically derived, from the very meaning of the expression lead to a semantic difficulty, S. Dialectic; Opposites; Consequence.

— Causal consequences. In the physical domain and natural experience, the effects predicted by the hypothesis are not attested, S. Refutation by the opposite. The refutation by an attested fact, different from the theoretically expected fact, is a kind of refutation from the absurd.

— Practical consequences. As soon as one turns from the scientifically established causal link to the “causal story” as constructed in a pragmatic argument, however, the speaker intervenes through his or her positive or negative valuation of the consequences. The consequence is then:

— Contrary to the intended goals, the effects of the proposed action are perverse; the measure is counterproductive, contrary to various interests.

— Inadmissible from the point of view of common sense, law, or morality, S. Apagogic; Ad Incommodum.

Pragmatic refutation by negative consequences is opposed to a measure by showing that it will have negative consequences unforeseen by the individual who proposes the measure, and that these drawbacks will prevail over any possible advantage.
The argument is strengthened  if the opponend can show that these  alledged negative consequences are not just  collateral effects, but are in fact  diametrically opposed to the expected positive effects:  the measure proposed to cure the patient will in fact strengthen her disease.

3. Demonstration by reduction to the absurd

Proof by the absurd, or by contradiction, is based on the principle of the excluded middle, according to which “A or not-A” is necessarily true. The rea- soning is based not on the proposition A that we want to prove, but on its negation, not-A.

The negation, not-A, is provisionally admitted and its consequences are de- duced; these consequences lead to statement A. But the conjunction “A and not-A” contravenes the principle of contradiction; thus, not-A is false, and A is necessarily true.

In the language of implication, we are in a situation where “A → non-A’. According to the principle of “one can deduce anything from the false”, this implication is true only if A is false.

It can be shown by reduction to the absurd that “the square root of 2 (the number whose square is 2, noted by the symbol √2) is not a rational number” (proposition A).

(1) Suppose that “the number corresponding to √2 is a rational number” (proposition not-A).
(2) By definition, a rational number can be written in the form of a fraction “p / q”, where p and q are prime numbers (a prime number can be divided only by itself and 1).

From this hypothesis, it can be deduced that both p and q are even; Therefore, they have 2 as a common divisor, which is contradictory to the initial hypothesis.

(8) Conclusion: hypothesis (1) is false, and √2 is not a rational number.

The demonstration by the absurd is an indirect method of demonstration. It has not been proved that A is true, but only that not-A is false. This reasoning is by no means permitted by all specialists, “if the classical mathematicians consider the proof by the absurd as valid, the intuitionists reject it: in order to prove a, they say, it is not enough to establish that not-(not-a)” (Vax 1982, Absurd). We see that the demonstrative character of a demonstration can be discussed.

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  Lat. absurdus, “absurd”. Argument ad absurdum, ab absurdo, ex absurdo; or reductio ad absurdum, “reduction to absurdity”, under different forms: reductio ad impossibile, “reduction to the impossible”; r. ad falsum, “r. to the false”; r. ad ridiculum, “r. to the ridicule”; r. ad incommodum; “r. to the undesirable”.
  

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