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

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

1. A claim, a proposal is put forward, as a working hypothesis or possibility.
2. Consequences are drawn from this proposition, regardless of whether they are causal or logical.
3. One of these consequences is deemed “absurd” based on certain criteria, cf. below
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 not-P has not been proven false.
— An argument to the absurd concludes that P is true because not-P has been proven false, and only one of P and not-P, can be true.
This corresponds to a case-by-case argument in a situation where the number of cases is reduced to two, either P is true or not-P is true; but not-P is false, so P is true, see 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 semantic consequences. Consequences derived, from the very meaning of an expression lead to a semantic difficulty, see dialectic; opposites.

— Causal consequences. In the physical domain and natural experience, the effects predicted by the hypothesis are not observed, see. refutation by the opposite. Refutation by an attested fact that differs from the theoretically expected fact, is a type of refutation from the absurd.

— Practical consequences. However, 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, see 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 opponent can show that these  alleged 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 reasoning 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 deduced. 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 as a fraction “p / q”, where p and q are prime numbers (a prime number can only be divided 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 contradicting the initial hypothesis.
(3) Conclusion: hypothesis (1) is false, and √2 is not a rational number.

The demonstration by the absurd is an indirect method of proof. It does not directly prove that A is true, only that not-A is false. Not all specialists agree with this reasoning, “while the classical mathematicians consider the proof by the absurd to be valid, the intuitionists reject it. They argue that in order to prove A, 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.