# Reductio ad absurdum

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* Add one: (2 × 3 × 5 × 7 × 11 × ··· × ''p''<sub>''N''</sub>) + 1. Call this number ''q''. | * Add one: (2 × 3 × 5 × 7 × 11 × ··· × ''p''<sub>''N''</sub>) + 1. Call this number ''q''. | ||

* Notice that ''q'' is not divisible by any of these primes up to ''p''<sub>''N''</sub>, since the remainder, when ''q'' is divided by any the primes, will always be 1. (For example, ''q'' divided by 2 gives the [[wikipedia:quotient|quotient]] 3 × 5 × 7 × 11 × ··· × ''p''<sub>''N''</sub>, with remainder 1.) | * Notice that ''q'' is not divisible by any of these primes up to ''p''<sub>''N''</sub>, since the remainder, when ''q'' is divided by any the primes, will always be 1. (For example, ''q'' divided by 2 gives the [[wikipedia:quotient|quotient]] 3 × 5 × 7 × 11 × ··· × ''p''<sub>''N''</sub>, with remainder 1.) | ||

− | * So ''q'' is either prime (''e''.''g''., 2×3×5 + 1 = 31) or it is divisible by a prime number which has not yet been listed (''e''.''g''., 2×3×5×7×11×13 + 1 = 30031 = 51×509). In either case, either ''q'' or one of [[wikipedia:Prime_factor|prime factors]] does not appear in our list of ''N'' prime numbers. | + | * So ''q'' is either prime (''e''.''g''., 2×3×5 + 1 = 31) or it is divisible by a prime number which has not yet been listed (''e''.''g''., 2×3×5×7×11×13 + 1 = 30031 = 51×509). In either case, either ''q'' or one of its [[wikipedia:Prime_factor|prime factors]] does not appear in our list of ''N'' prime numbers. |

* Thus we have found a prime number not appearing in our list of ''N'' prime numbers and thus there are at least ''N'' + 1 prime numbers. This is a contradiction. | * Thus we have found a prime number not appearing in our list of ''N'' prime numbers and thus there are at least ''N'' + 1 prime numbers. This is a contradiction. | ||

* Therefore, we have to reject our original assumption. There must be infinitely many prime numbers. | * Therefore, we have to reject our original assumption. There must be infinitely many prime numbers. |

## Revision as of 16:40, 14 October 2010

**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 and call this number
*N*. - We can therefore list all
*N*prime numbers: 2, 3, 5, ...,*p*_{N}. - Consider the product of all
*N*prime numbers: 2 × 3 × 5 × 7 × 11 × ··· ×*p*_{N}_{}*.* - Add one: (2 × 3 × 5 × 7 × 11 × ··· ×
*p*_{N}) + 1. Call this number*q*. - Notice that
*q*is not divisible by any of these primes up to*p*_{N}, since the remainder, when*q*is divided by any the primes, will always be 1. (For example,*q*divided by 2 gives the quotient 3 × 5 × 7 × 11 × ··· ×*p*_{N}, with remainder 1.) - So
*q*is either prime (*e*.*g*., 2×3×5 + 1 = 31) or it is divisible by a prime number which has not yet been listed (*e*.*g*., 2×3×5×7×11×13 + 1 = 30031 = 51×509). In either case, either*q*or one of its prime factors does not appear in our list of*N*prime numbers. - Thus we have found a prime number not appearing in our list of
*N*prime numbers and thus there are at least*N*+ 1 prime numbers. 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 (as its possible, however unlikely that life arose by something other than purely natural means) the absurdity is not shown. The assertion that its absurd is only used to come to the conclusion that God did it. The person that would be making this argument does not know how life arose, so to assume anything other than a deity seems absurd. (See: God of the gaps)