Russell's paradox

This article is about a paradox of self-reference. For an unrelated paradox in the theory of logical conditionals with a similar name, introduced by Lewis Carroll, see the Barbershop paradox.

The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible.

Paradox

Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:

1. shaving himself; or
2. going to the barber.

Another way to state this is that "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves."

From this, asking the question "Who shaves the barber?" results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities are valid: they both result in the barber shaving himself, but he cannot do this because he only shaves those men "who do not shave themselves".

History

This paradox is often attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to him as an alternate form of Russell's paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber's paradox was an instance of his own:

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This point is elaborated further under Applied versions of Russell's paradox.

In Prolog

In Prolog, one aspect of the barber paradox can be expressed by a self-referencing clause:

shaves(barber, X) :- male(X), not shaves(X,X).
male(barber).

where negation as failure is assumed. If we apply the stratification test known from Datalog, the predicate shaves is exposed as unstratifiable since it is defined recursively over its negation.

In first-order logic

$(\mathrm{\exists }x)(\text{man}(x)\wedge (\mathrm{\forall }y)(\text{man}(y)\to (\text{shaves}(x,y)\leftrightarrow \mathrm{\neg }\text{shaves}(y,y))))$

This sentence is unsatisfiable (a contradiction) because of the universal quantifier $(\mathrm{\forall })$. The universal quantifier y will include every single element in the domain, including our infamous barber x. So when the value x is assigned to y, the sentence can be rewritten to $\text{shaves}(x,x)\leftrightarrow \mathrm{\neg }\text{shaves}(x,x)$, which simplifies to $\text{shaves}(x,x)\wedge \mathrm{\neg }\text{shaves}(x,x)$, a contradiction.

In literature

In his book Alice in Puzzleland, the logician Raymond Smullyan had the character Humpty Dumpty explain the apparent paradox to Alice. Smullyan argues that the paradox is akin to the statement "I know a man who is both five feet tall and six feet tall," in effect claiming that the "paradox" is merely a contradiction, not a true paradox at all, as the two axioms above are mutually exclusive.

A paradox is supposed to arise from plausible and apparently consistent statements; Smullyan suggests that the "rule" the barber is supposed to be following is too absurd to seem plausible.

The paradox is also mentioned several times in David Foster Wallace's first novel, The Broom of the System as well as The Information, by James Gleick.

Multiple barbers

If the paradox is altered so that there may be multiple barbers in the town, then the paradox may or may not be resolved, depending on the exact phrasing of the initial rules.

If the initial rules state that every man in town must keep himself clean-shaven, either by

1. Shaving himself, or
2. going to a barber.

(but not both at once), then the paradox is solved. Each barber can be shaved by another barber.

However, if the initial rules describe the responsibilities of the barbers rather than the town's residents in general, then the paradox remains. In this version, the rules state that each barber must shave everyone in town who does not shave himself (and no one else). If Barber A asks Barber B to shave his beard, then Barber A counts as "a person who does not shave himself". But because of this classification, Barber A must shave himself, rather than let Barber B do it for him. However, if Barber A is shaving himself, then he must not shave himself. Either way, Barber A is stuck. Other barbers face the same problem.

Non-paradoxical variations

A modified version of the barber paradox is frequently encountered in the form of a brainteaser puzzle or joke. The joke is phrased nearly identically to the standard paradox, but omitting a detail that allows an answer to escape the paradox entirely. For example, the puzzle can be stated as occurring in a small town whose barber claims: I shave all and only the men in our town who do not shave themselves. This version identifies the sex of the clients, but omits the sex of the barber, so a simple solution is that the barber is a woman. The barber's claim applies to only "men in our town," so there is no paradox if the barber is a woman (or a gorilla, or a child, or a man from some other town—or anything other than a "man in our town"). Such a variation is not considered to be a paradox at all: the true barber paradox requires the contradiction arising from the situation where the barber's claim applies to himself.

Notice that the paradox still occurs if we claim that the barber is a man in our town with a beard. In this case, the barber does not shave himself (because he has a beard); but then according to his claim (that he shaves all men who do not shave themselves), he must shave himself.

In a similar way, the paradox still occurs if the barber is a man in our town who cannot grow a beard. Once again, he does not shave himself (because he has no hair on his face), but that implies that he does shave himself.

See also

• Halting problem
• List of paradoxes

External links

• Proposition of the Barber's Paradox
• Joyce, Helen. "Mathematical mysteries: The Barber's Paradox." Plus, May 2002.
• Edsger Dijkstra's take on the problem
• The Monist, Vol. 29, No. 3, JULY, 1919, THE PHILOSOPHY OF LOGICAL ATOMISM, page 354