# Difference between revisions of "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.

## 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:

1. Create an initial assumption
2. Follow a series of axiomatically valid steps
4. Therefore, the initial assumption is incorrect

### Proof of Proposition P by Contradiction

1. Suppose ~P.
2. ...
3. Therefore, Q.
4. ...
5. Therefore, ~Q
6. Hence, Q and ~Q, a contradiction
7. Thus, P

### Example

An example of this is Euclid's proof of the infinitude of the primes:

1. Assume there are finitely many primes
2. 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
3. By the fundamental theorem of arithmetic, N+1 has a prime factorization. But N+1 is not divisible by any of the previous primes
4. Since N+1 is composite, there must be a prime missing from our set of primes. But this set contains all primes
5. Therefore, our initial assumption ("there are finitely many primes") is invalid