Paradox of the tattoo artist

Stella asked:

Here is a puzzle that requires some ‘conceptual analysis’. You really need to think about the meaning of the words involved and the way this guy Eugene is described!! Good luck!!

RICK: Let me tell you about my hometown, Tattoopolis, and my favorite tattoo artist there, Eugene. As of today, everyone in my hometown has exactly one tattoo (though they could have more in the future). That of course makes it an interesting town. But Eugene is even more interesting! Eugene tattoos all and only those people in the town who do not give tattoos to themselves! That’s right: he tattoos all people in the town who don’t give tattoos to themselves, AND he only tattoos people in the town who don’t give tattoos to themselves.

SLICK: Wow, that’s quite a story there, Rick. Too bad it’s false. I know for sure that there’s no such person as Eugene.

Slick is right. There is no such person as Eugene. Slick knows this even though he has never been to Tattoopolis nor has he ever talked to anyone from Tattoopolis (other than Rick). As a matter of fact, Slick doesn’t need to know anything else about Tattoopolis or who lives there to know that there is no such person as Eugene.

So how does he know that Eugene does not exist? (Hint: It has to do with the concept that supposedly applies to Eugene).

Answer by Geoffrey Klempner

The short answer to your question is that the description of Eugene is self-contradictory. Self-contradictory entities (like round squares) cannot exist.

That’s it.

However, there is a longer and more interesting answer. The story of Slick and Eugene is in fact a version of a well known paradox known as the ‘Barber paradox’. There is a Barber in [whichever city you like] who shaves all and only those men who do not shave themselves. Does he shave himself or not? (I suspect that the barber has been changed to a tattoo artist to encourage gender equality!)

The Barber (or Tattoo) paradox isn’t really a paradox. There’s no problem to solve, once we see that it has a solution, albeit one that involves rejecting a question. We reject the question whether the barber shaves himself or whether the tattooist tattoos him/ herself, for the same reason as I would reject the question whether or not I have stopped beating my wife. I haven’t stopped beating her and I haven’t not stopped beating her, because I have never beaten her. The barber does not shave himself and does not not shave himself because, logically, there can be no such barber.

If this is all paradoxes amounted to, then they wouldn’t cause such headaches for philosophers. However, there are other paradoxes which have caused great consternation and for which there is no agreed solution.

One paradox which is very similar in form to the Barber paradox is Russell’s Paradox, originally discovered by Gottlob Frege. Consider classes which are not members of themselves. The class of carrots, for example, isn’t a carrot. On the other hand the class of abstract objects IS an abstract object.

Let’s take the class of ALL the classes, like the class of carrots, which are not members of themselves. Is it a member of itself or not? You already know the answer, if you’ve followed the explanation of the Barber paradox. There is, logically can be, no such class.

But that is genuinely paradoxical. There seems no logical reason why we can’t form a class of all the classes which are not members of themselves! Russell spent years on this — according to his Autobiography it drove him to the brink of despair and ruined his marriage — finally coming up with a solution that he was not fully happy with, because it involves a rule restricting the formation of classes whose only real motivation is that it avoids the paradox. Other mathematicians and philosophers have proposed their own solutions, which are no less arbitrary.


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