The Barber Paradox

By Linda Shaver

 A brainteaser:

In a particular town, there’s a particular barbershop with a peculiar sign in the window that reads:

“This barber shaves all and only those men of the town who do not shave themselves.”

According to the sign in the window, does the barber shave himself?

(Those of you who are very adept at brainteasers might say, “Ha!  You never said the barber was ‘of the town’!  Maybe he’s from another town!  So there!”  Fair enough.  Were it not for my use of the word “himself” in the question, you might have also pointed out that the barber could be a woman, since not every person who shaves others is male.  Indeed, the Shaver who wrote this article is a woman.  But for our purposes, this particular barber is a man and is of this town.)

Does the barber shave himself?

Let’s say that he does.  But the sign claims that he only shaves those men who do not shave themselves.  We just said that he does shave himself, which means he doesn’t fit the description of the only type of person he shaves, so he cannot shave himself.

So, he doesn’t shave himself.  But now he fits the description of a man who does not shave himself, and since his sign claims that he shaves every man who does not shave himself, he must be shaved by him.

In other words: If he shaves himself, then he must not shave himself.  If he doesn’t shave himself, then he must shave himself.  It’s a paradox.

How does this happen?  Using notation similar to that found in Language Proof and Logic (but adapted to fit the constraints of the available font), the barber’s sign translates as:

Ax Ey [[M(x) ^ B(y)]  → [ ¬Shaves (x,x) ↔ Shaves (y,x)]]

where A represents the universal quantifier, E represents the existential quantifier, M(x) stands for “x is a man of the town,” and B(y) stands for “y is the barber.”  In English, this sentence is read as, “For every x, there exists a y such that, if x is a man of the town and y is the barber, then it is not the case that a man of the town shaves himself if and only if the barber shaves x.” Indeed, this is what the original sign says, albeit less abstrusely.

When the sentence is universally instantiated over y, it becomes:

Ey [[M(y) ^ B(y)]  → [ ¬Shaves (y,y) ↔ Shaves (y,y)]]

“There exists a y such that, if y is a man of the town and is the barber, then it is not the case that he shaves himself if and only if he shaves himself.”  Since for our purposes we’ve established that the barber is a man of the town, the conditional is satisfied, and (after a bit of “hand waving” to get rid of the existential quantifier) we’re left with:

¬Shaves (y,y) ↔ Shaves (y,y)

“It is not the case that the barber shaves himself if and only if he shaves himself.”  The paradox.

So, what is the barber to do?  He should probably reword his sign.

(For more information on the Barber Paradox, consult pages 333-4 in LPL.)