This last demonstration provides an example of an "indirect" method of proof. One makes an assumption in order to prove it couldn't be true!

In our example, we assumed that our unsound but valid argument A had no false premises. We showed that this would mean that A is sound. But, from the way we defined it, A is not sound.

The upshot is that the assumption in red must be mistaken. We'd better take it back if we don't want to contradict ourselves!

By the way, this contradiction is sometimes called an "absurdity". And this indirect method of proof is sometimes called "reductio ad absurdum". It reduces the mistaken assumption to an absurdity!

So, this method has a funny name and is sometimes a little hard to get used to, but it can be a powerful technique in thinking. Here's the synopsis:

To use the indirect or "reductio ad absurdum" method to prove some proposition P, start by assuming that P is false and finish by showing that this assumption leads to contradiction. (Thus, because assuming P false turns out to be absurd, P must be true.)

Indirect Proofs

Review of this Tutorial
Exercises on informal proofs
Back to chapter one