# Reductio ad absurdum

**Reductio ad absurdum** is a type of logical argument where one assumes a claim for the sake of argument, arrives at an "absurd" result (often a contradiction), and then concludes that the original assumption must have been wrong, since it led to this absurd result.

Note that this is a logically valid technique. It is a form of *modus tolens*, an inference rule which takes this form:

- If
**P**then**Q**. -
**Q**is false. - Therefore
**P**is false.

More formally, a *reductio ad absurdum* argument typically takes the form:

- Assume
**P**. - This implies
**Q**. - It also implies
**R**. - But
**Q**and**R**are contradictory (**Q**iff not**R**). - Therefore
**P**is false.

Here's a real example (from number theory) of this type of argument in action:

- Assume there are finitely many prime numbers.
- Then there is a largest prime number. Call it
*p*. - Consider the product of all the prime numbers up to this largest prime: 2 × 3 × 5 × 7 × 11 × ... ×
*p*. - Add one: (2 × 3 × 5 × 7 × 11 × ... ×
*p*) + 1. Call this number*N*. - Notice that
*N*is not divisible by any of the primes up to*p*, since the remainder, when*N*is divided by any the primes, will always be 1. (For example,*N*divided by 2 gives the quotient 3 × 5 × 7 × 11 × ... ×*p*, with remainder 1.) - So
*N*is prime. And clearly,*N*is larger than*p*. - Thus we have found a prime number larger than the supposed "largest prime". This is a contradiction.
- Therefore, we have to reject our original assumption. There must be infinitely many prime numbers.

## Counter-apologetics

See Can God create a rock so heavy that he can't lift it? for an example in the context of counter-apologetics (the claim being assumed is that God is all-powerful).

The problem with this type of argument is that the "absurdity" one reaches must actually be a logical contradiction in order for the argument to be valid. If the conclusion is simply unlikely, then the argument doesn't necessarily work. For example:

- If God doesn't exist, then life arose by purely natural means.
- This is absurd (read: very, very unlikely).
- Therefore, God exists.

Well... no. Ignoring the fact that the premise is faulty,