Barber 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 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. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself.

Paradox

The barber is the "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself?

Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be a barber. If the barber does not shave himself then he fits into the group of people who would be shaved by the barber (and, so, as the barber he needs to shave himself).

Criticism

Despite its popular name, however, Barber paradox is not really a paradox in the true sense of this word. A man who shaves exactly those men who do not shave themselves simply cannot and does not exist, and there are virtually no reasons to expect the opposite. This is in contrast with the set of all sets that do not contain themselves (from Russell's paradox), whose existence cannot be painlessly dismissed as it follows from the very intuitive and widely relied upon axioms of naive set theory.

History

This paradox is often attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to him as an alternative 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:

That contradiction [Russell's paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.

— Bertrand Russell, The Philosophy of Logical Atomism

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

In first-order logic

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

This sentence is unsatisfiable (a contradiction) because of the universal quantifier $(\forall)$. 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 $\text{shaves}(x,x)\leftrightarrow \neg \text{shaves}(x,x)$, which simplifies to $a \leftrightarrow \neg a$, a contradiction.

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 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 brain teaser 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

• Cantor's theorem
• Halting problem
• List of paradoxes
• Double bind

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