One way to establish a proposition is to construct a sound argument which has that proposition as its conclusion.

A different way to establish a proposition is to begin by assuming the opposite -- i.e., the negation -- of what you want to establish, and then deriving a falsehood from this assumption. This shows that the original assumption must be false, and hence that its opposite -- which is the claim that you wanted to establish -- is true. When employing this mode of argument you reduce the assumption to absurdity -- hence the name `reductio ad absurdum.'

For example, suppose that you want to prove that there is no town with a barber who shaves all and only those men who do not shave themselves. One way to do this is by reductio: assume to start that there is a town such which contains a barber who shaves every person who does not shave himself. That barber must either shave himself or not shave himself. Suppose that he shaves himself; then he does not shave himself, since he shaves only those people who do not shave themselves. Suppose that he does not shave himself; then he must shave himself, since he shaves everyone who does not shave himself. So, beginning with the assumption that there is a town with a barber who shaves all and only those people who do not shave themselves, we have derived the absurd conclusion that the barber both shaves himself and does not shave himself. So our original assumption has been reduced to absurdity, and must be false. So the opposite of our original assumption -- namely, that there is no town with a barber who shaves all and only those men who do not shave themselves -- must be true. And this is what we were trying to prove.