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.
The Liar paradox arises when we ask ourselves whether the following sentence, called a “Liar sentence”, is true or not:
This sentence is not true.
The sentence just above this line says of itself, so to speak, that it is not true. If we suppose that the sentence is true, then what the sentence says (that it is not true) must be the case and, thus, the sentence is not true. But if we suppose that the sentence is not true, then what the sentence says (again, that it is not true) turns out to be the case and, hence, the sentence is true after all.
The Liar paradox was already known by the ancient Greeks and, after more than 2000 years, is yet to be solved.