Reductio ad absurdum

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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,

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