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What is the exact difference between reductio ad absurdum and proof by contradiction?

Wikipedia used to state that:

Reductio ad absurdum (Latin: "reduction to the absurd") is a form of argument in which a proposition is disproven by following its implications logically to an absurd consequence.

When I read this, I instantly thought, "Ah, that's proof by contradiction...". However, it continues like this:

A common species of reductio ad absurdum is proof by contradiction (also called indirect proof), where a proposition is proved true by proving that it is impossible for it to be false.

So, I suppose there is some subtle difference between them that isn't clearly explained in the article. What exactly is that subtle difference? How are the two strategies related, and how do they differ?

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    This citation from Wikipedia serves, I fear, to support a case that Wikipedia is not an especially good resource for philosophy.
    – vanden
    Jun 21, 2011 at 5:13
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    @vanden: That's a pretty good call, and I think it's strange because logic is typically very well covered.
    – boehj
    Jun 21, 2011 at 8:15
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    Related post on the math site: math.stackexchange.com/questions/240/…, and see also mathoverflow.net/questions/12342/…
    – JDH
    Jul 4, 2011 at 12:54
  • This is an old question, and Wikipedia has changed quite a lot since you wrote it. It's worth pointing out that the lead part of the Wikipedia page now explains this fairly well: as an informal method of argument, the "absurdity" to which one reduces might not be a logical contradiction but might instead be something at odds with observed reality, or simply intuitively unreasonable.
    – N. Virgo
    Dec 9, 2015 at 11:11
  • wikipedia like SE can grow over time and has fixed some of this language (still a great question since the previous language is a common misunderstanding).
    – virmaior
    Aug 6, 2016 at 1:18

2 Answers 2

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There is some instability in the terminology here. Many authors use Reductio Ad Absurdum (RAA) as meaning the same as proof by contradiction and indirect proof. More careful authors distinguish them, taking both RAA and indirect proof to be a species of proof by contradiction.

In what follows, I use P and Q for propositional meta-variables, ∧ for conjunction, ∨ for disjunction and ¬ for negations. (I do wish for LaTeX!)

RAA proceeds by assuming some proposition P, on that basis deriving some contradiction such as Q ∧ ¬Q, and, having reduced P to absurdity, concluding ¬P. In the context of a natural deduction proof system for logic with introduction and elimination rules for each connective, this is ¬-Introduction.

Indirect proof is the very similar method of proof whereby you assume ¬P, derive a contradiction such as Q ∧ ¬Q and then conclude that P. In the sort of natural deduction presentation just mentioned, this is the rule of ¬-elimination.

These two principles are very much worth distinguishing. If you take a classical propositional logic natural deduction proof system with introduction and elimination rules for each connective and remove ¬-elimination, the result is intuitionistic logic. This logic is more often thought of as being characterized by the denial of the law of excluded middle (P ∨ ¬P), but the two characterizations are equivalent for propositional logic.

Intuitionistic logic is one of the best studied and oldest non-classical logics, and one which plays a prominent role in many debates in metaphysics.

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    @loudandclear: Not a whole lot hangs on it, but I really did intentionally opt for '~' rather than '¬' in this case. While there are plenty of conventions, here, many people do use '~' for classical (truth-functional) negations and '¬' for intuitionistic (and other constructive logic's) negation. In general, I'm going to use the symbols easily available on the keyboard; if you and others want to change them, that's fine :)
    – vanden
    Jun 21, 2011 at 16:26
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    Interesting. I approved the edit without knowing that distinction. I'm hardly a student of formal logic (quite the opposite, in fact), but that's good to know.
    – Cody Gray - on strike
    Jul 4, 2011 at 12:07
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    In math logic circles, it is extremely common to use ¬ for negation in the context of classical logic, so this usage is certainly not incorrect; indeed, in my experience it is much more commonly used than ~. But of course, the need to distinguish the symbols arises only in non-classical logic, where I imagine that such conventions exist.
    – JDH
    Jul 7, 2011 at 11:43
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    So, in short, in RAA you assume P and conclude not P, and in Indirect Proof you assume not P and conclude P, and both of these are methods of doing a Proof by Contradiction?
    – Pro Q
    Nov 18, 2018 at 1:54
  • @ProQ came to the comments to ask exactly this, that's what it seems like
    – Joseph Garvin
    Jan 28 at 21:02
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There are different types of absurd consequence (absurdities). The two main ones being pairs of contradictory and contrary statements. Proof by contradiction meets the first type of absurdity. See the Square of Opposition from syllogistic logic for more on this.

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