Reductio ad Absurdum

§4.7

“Reductio ad absurdum” (“reductio” for short) is Latin for “reduction to absurdity”. It refers to the strategy of proving a proposition false by deducing an absurd conclusion (often a contradiction) from it in conjunction with other known premises. Such arguments are common in math. They were used to prove that the square root of two is not a rational number (and thus that there are such things as irrational numbers) and that there is no largest prime number. Here’s a non-mathematical example of a reductio.

There cannot be a chess-playing computer program good enough to win every game against any opponent, regardless of whether it plays as white or black. If there were such a program, it could be pitted against itself, and both sides would have to win. But it’s impossible for both sides to win in a chess game, therefore there cannot be a chess-playing program such as the one described.13FTNOTE TBA

And here is an example from mathematics, first in paragraph form and then laid out:

Suppose that the set of prime numbers is finite. If so, it must contain a largest member—call it L. We can then multiply all the prime numbers together. Call the result P. P will be divisible by each of the prime numbers, so P+1, won’t be divisible by any prime number. Therefore, it won’t be divisible by any number, which means that it will be prime. But P+1 is larger than L. So, L isn’t the largest prime number. But this is a contradiction, so the assumption that lead to it must be false, and the set of prime numbers must be infinite.

Map 42:

In assessing a reductio ad absurdum you need to determine two things: (1) whether the supposedly absurd conclusion is really false, and (2) whether it really follows necessarily from the proposition that the argument sets out to disprove. With regard to the first of these issues, the strongest reductios deduce contradictions, which it is easy to see cannot be true, but in other cases it is less certain that the “absurd” conclusion is false. With regard to the second issue, you assess a reductio in the same way that you would access any other deductive argument: you determine whether the inference is valid and how certain each of the premises is other than the one that has been assumed for the sake of refuting it. In order for the argument to prove that the assumed premise is false, all the other premises must be certain, and it must be certain that the “absurd” conclusion is false. However, even if this is not the case, the reductio can still give you strong reason to disbelieve the assumed premise if all of the other premises and the falsehood of the absurd conclusion all have a much higher epistemic status than the assumption that the argument is trying to show to be false.