Reductio Ad Absurdum
Reducto ad absurdum is Latin for "reduction to the absurd." It is a form of argument that seeks to disprove a proposition by showing that that proposition inevitably leads to an absurdity, or of proving a proposition by assuming its negation and then showing that that assumption inevitably leads to an absurdity.[1] One use is in mathematics, as will be described in greater detail below. Reductio ad absurdum should not be confused with the straw man fallacy or the slippery slope fallacy.
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In Mathematics
Reductio ad absurdum, also called "proof by contradiction" or "proof by assuming the opposite," is a method of mathematical proof. It involves assuming the opposite of what one is trying to prove, and showing that this would lead to a contradiction. It works by the law of the excluded middle. The proof typically follows this structure:
- Create an initial assumption
- Follow a series of axiomatically valid steps
- Reach a contradiction
- Therefore, the initial assumption is incorrect
Proof of Proposition P by Contradiction
- Suppose ~P.
- ...
- Therefore, Q.
- ...
- Therefore, ~Q
- Hence, Q and ~Q, a contradiction
- Thus, P
Example
An example of this is Euclid's proof of the infinitude of the primes:
- Assume there are finitely many primes
- Take the product of all primes and call it N. Since N+1 is not in our finite set of primes, it must be composite
- By the fundamental theorem of arithmetic, N+1 has a prime factorization. But N+1 is not divisible by any of the previous primes
- Since N+1 is composite, there must be a prime missing from our set of primes. But this set contains all primes
- Therefore, our initial assumption ("there are finitely many primes") is invalid
See also
- Stolen concept, a fallacy that can be exposed using a reductio ad absurdum