The paradox of the Barber was first introduced by the British philosopher Bertrand Russell (1872-1970). Russell asks us to consider a village with just one barber who has some restrictions on which people he shaves: he must shave all and only those villagers who do not shave themselves.
The paradox arises when we ask ourselves whether the Barber shaves himself or not. If he does not, then he is one of the villagers who do not shave themselves and, consequently, he must shave himself. But, if he shaves himself then he must be one of the villagers who do not shave themselves, for he only shaves such villagers. Thus, if he does not shave himself, he must do it; and if he shaves himself, he must not do it.
This paradox is usually taken to show that such a barber cannot exist.