# 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 that usually takes one of two forms:

1. Self contradiction:

- Assume that
**P**is true. - Deduce that
**P**must also be false. - Therefore,
**P**is false.

2. Deducing two mutually contradictory statements:

- Assume
**P**is true. - From this assumption, deduce that
**Q**is true. - Also deduce that
**Q**is false. - Thus,
**P**implies**Q**and**not Q**or, more simply,**P**implies a falsehood. - Use
*modus tollens*to conclude that**P**itself must be false.

An example (from number theory) of the first form of this type of argument is:

- 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.

An example of the second form of this type of argument comes from Euclid's *Elements* which states that if two circles cut each other, they must have different centers:

- Assume that if two circles cut each other, they have the same center
*Z*. - The length of a ray
*ZC*to one of the cuts*C*(a point lying on both circles) must therefore equal the length of a ray*ZA*to a point*A*on the first circle that is not also on the second circle. - The line forming this ray, however, must also cut the second circle at a point
*B*and the length of*ZB*therefore be different from the length of*ZA*. - But, be definition,
*ZB*equals the length of the ray*ZC*. - Things which are equal to a common thing must therefore be equal to each other. Thus, the length of
*ZA*must equal the length of*ZB*. - We have therefore deduced that the lengths of
*ZA*and*ZB*must be both equal and unequal. - Therefore, the original assumption is false: the circles must have different centers.

## 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)