The barber of Sevilla, as reformulated by Russell, as reformulated by Wikipedia:

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.

Under this scenario, we can ask the following question: Does the barber shave himself?

Well, since the barber is a man and thus must keep himself clean-shaven, then either he shaves himself or someone else does it for him. Since he’s the only barber, he must shave himself.

But! The barber shaves “all and only those men in town who do not shave themselves”. So from this point of view, the barber should not shave himself; however, if he would not shave himself, the only barber (himself) would do it for him. …eh?

Something’s wrong – looks like a “discontinuity” due to contradictions in the premises. How to resolve this?

We could assign priorities to the premises:

• If the rule “all men are clean-shaven” has highest priority, then the barber must shave himself: if the barber does not shave himself, his beard will grow and thus condition “everyone is clean-shaven” will not hold. Thus in this case, he simply must shave himself.
• But if the barber does shave himself, then he breaks the rule of “shaving all and only those men who do not shave themselves”. So if this rule has highest priority, then the barber must not shave himself.

So, we might have a way out by assigning a total order for the premises, such as “rules are given in descending order of priority”.

But, since Mr. Russell has presented us with this neat paradox, and taking into account his supreme reputation and cunning in the ways of logic, good sportsmanship dictates that we should handle this paradox somehow within logic. Otherwise, Mr. Russell might appear and beat the shit out of us with logical atomism or some other strange contraption.

“Damn right I would”

So what can we do?

Let’s consider a case where the prioritized premises (rules) have equal priority; we can’t favor one over the other. Next, take a trip to Zen-Buddhist koans and get some inspiration from the world of quantum weirdness and Schrödinger’s feline exploitations, and re-express the core of the original “paradox” in an equivalent form: each man in the village shaves themselves, except for the barber who both shaves and not-shaves himself.

You can’t shave and not-shave! (Not at at the same time at least. Exercise for reader: Or can you? What would it look like?)

Now, if I were a computer, “shave and not-shave” as a condition would not compute and therefore I would not accept this paradox as input. It’s invalid input. I’d barf up an error message, insult the user, and go on with my computations.

Thusly we can conclude: ?SYNTAX ERROR, Mr. Bertrand Russell!

Update: Thanks to Carl “Benson Carl” Benson for telling me about this paradox.

Note to reader: You can achieve the same result in a few lines by using first-order logic. But then we wouldn’t have this wonderful blog post!