# 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.

## Contents

## 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*