The task of this discussion is to explain the questionable argument of the reductio and to illustrate the scope of its applications. As noted above, this type of proof of a thesis by reductio argumentation, which draws a contradiction from its negation, is called indirect proof in mathematics. (For historical context, see T. L. Heath, A History of Greek Mathematics [Oxford: Clarendon Press, 1921].) In Principia Mathematica, Whitehead and Russell characterize the principle of «reductio ad absurdum» as equivalent to the formula (~p →p) →p of propositional logic. But this view is idiosyncratic. Elsewhere, the principle is almost universally regarded as a mode of reasoning rather than as a specific thesis of propositional logic. The propositional reductio is based on the following argument: One last point. The contrast between reductio and impossible reasoning teaches an interesting lesson. In both cases, we start from a situation of exactly the same basic format, namely a conflict of contradiction between a hypothesis of hypotheses and various facts that we already know. The difference lies entirely in pragmatic considerations, in what we are trying to achieve. In one case (reductio), we try to refute and refute these hypotheses in order to establish their negation, and in the other case (by impossible), we try to establish an implication – validate a condition. So the difference is fundamentally not in the nature of the conclusion in question, but only in what we are trying to achieve by its means.

So the difference is not so much theoretical as functional – it is a pragmatic difference in objectives. What we have here are consequences that are absurd in the sense that they are obviously false and even a little ridiculous. Although it departs from what is strictly interpreted – conditions with contradictory conclusions – temporary – such a thing is also characterized as a weakened mode of reduction. But while all three cases fall into the realm of the term as it is commonly used, logicians and mathematicians usually have the first and strongest of them in mind. The reductio ad absurdum is also called «reduction to absurdity». It is a question of characterizing a counter-argument in such a way that it seems ridiculous or the consequences of the position seem ridiculous. This can be ridiculous in the sense that the argument seems ridiculous, or ridiculous in the sense that no reasonable person would take such a position. I note this here because a good reductio ad absurdum may require a lot of critical analysis to maintain, so much so that it is often better to simply ignore such arguments.

The mathematical school of so-called intuitionism has adopted a clear line regarding the limitation of reductio`s reasoning in order to prove existence. The only valid way to prove existence, they argue, is to provide a concrete example: a general argument of principle is not acceptable here. In particular, this means that (∃x)Fx cannot be determined by deriving an absurdity from (∀x)~Fx. As a result, intuitionists would not allow us to infer the existence of the invertebrate ancestors of Homo sapiens from the obvious absurdity of the assumption that humans are vertebrates. They would say that in such cases, where we are completely ignorant of the people involved, we are unable to maintain their existence. Similarly, reductio ad absurdum can refer to a type of argument in which something is proven by showing that the opposite is not true. Also known as indirect proof, contradictory proof, and classical reductio ad absurdum. As Morrow and Weston point out in A Workbook for Arguments (2015), arguments developed by reductio ad absurdum are often used to prove mathematical theorems. Mathematicians «often call these arguments `proofs by contradiction.` They use this name because Reductio`s mathematical arguments lead to contradictions – such as the claim that N is both the largest prime number and not. Since contradictions cannot be true, they provide very strong reductio arguments. Aristotle clarified the connection between contradiction and falsehood in his principle of non-contradiction, which states that a sentence cannot be both true and false. [13] [14] That is, a Q phrase {displaystyle Q} and its negation ¬ Q {displaystyle lnot Q} (not-Q) cannot both be true.