Paradox of the barber

The paradox of the barber is an illustration with didactic goal of the Paradoxe of Russell, allotted to Bertrand Russell him even. One thus should not give an excessive importance to this “Paradoxe”, that the Logicien E.W. Beth qualifies “discrepancy claimed” or of “pseudo-discrepancy”.

Statement of the paradox of the barber

One can state the paradox as follows:

The municipal council of a village adopts an ordinance which enjoint with its barber (male) to shave all the male inhabitants of the village who do not shave themselves and only those.

The barber, who is well an inhabitant of the village, could not comply with this rule because:

  • If it shaves itself, it enfreint the rule, because the barber can shave only the men who do not shave themselves;
  • If it does not shave itself (which it is made shave or that it preserves the beard), it is in wrong also, because it with the responsibility of shave the men who do not shave themselves.

This rule is thus inapplicable. Does it act for all that a paradox? There is no reason to think that a council of village or any other authority cannot return an ordinance absurdity. In fact, far from being a logical discrepancy, this “parodoxe” watch simply that a barber complying with this rule cannot exist. It is about an illustration of what, if R is an unspecified binary relation (in fact “… shaves…”), the following statement, writing in formal language:

¬ ∃ there ∀ X ( there R X ⇔ ¬ X R X )

is a universally valid formula of the Calcul of the predicates first order. One will refer to the article on the Paradoxe of Russell to see why that can lead, in the case of the relation of membership in a too naive set theory, with a true discrepancy, i.e. with a contradiction shown in the theory.

The paradox of the barber constitutes a particularly simple illustration of the diagonal Argument of Cantor, which does not mean only this last is reduced to this one. As it applies in fact to any relation (binary), one can give of them, with more or less happiness, multiple alternatives. Let us quote this one, due to Martin Gardner: is it logically possible to write an encyclopedia which indexes all the encyclopedias not indexing themselves and only those? The answer is not, since this encyclopedia can neither be indexed, nor not to index itself.

Related articles

Sources

  • E.W. Beth, logical bases of mathematics - Gauthier-Villars (Paris)/E. Nauwelaerts (Leuwen) 1950.

  • Martin Gardner, magic of the paradoxes - For Science 1985.
  • Nicholas Falletta, the book of the paradoxes - Belfond/sciences, 1988 ISBN 2.7144.1789.2
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