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. 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.
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
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
- Stolen concept, a fallacy that can be exposed using a reductio ad absurdum