Bertrand Russell’s barber paradox

A barber shaves all the men in a village who do not shave themselves.
So who shaves the barber?
If the barber shaves himself, then he belongs to the subset of men who are not shaved by the barber.
If he does not shave himself, then the barber does.
Which means…


What?
As nonsensical as any result seems, this is the kind of thing that theoretical mathemeticans, philosophers and others puzzle over. It gets at the heart of the question:
Does everything exist? If not “here,” then somewhere?
If not, then why does only whatever is out there exist? What governs the circumstances under which whatever is, is?

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  1. #1 by mxracer652 on July 6, 2007 - 8:58 am

    Someone get the water pistol, Kevin’s getting existential again.

  2. #2 by mxracer652 on July 6, 2007 - 8:58 am

    Someone get the water pistol, Kevin’s getting existential again.

  3. #3 by Dunc on July 6, 2007 - 9:28 am

    The barber is a woman. ;)
    Or, less facetiously, the barber is not operating as a barber when he shaves himself. The term “barber” denotes a relationship between two people – the person doing the shaving, and the person being shaved. He’s only a barber when he’s shaving clients. When he shaves himself, he’s just a man.
    How’s that?

  4. #4 by Taylor Murphy on July 6, 2007 - 10:04 am

    If the universe is expanding such that something is moving away from us at faster than the speed of light, and it makes no contact with us whatsoever, does it exist? hurf

  5. #5 by Flaky on July 6, 2007 - 10:05 am

    Logician to the rescue!
    A barber shaves all the men in a village who do not shave themselves.

    If the barber shaves himself, then he belongs to the subset of men who are not shaved by the barber.

    Ah, but the first assertion does not state that all of the men who shave themselves are not shaved by the barber, making the inference unsound. (Of course one could restate the first assertions so that the groups may not overlap, which means that either the barber doesn’t need shaving or no such village exists if barbers always need shaving.)

  6. #6 by Alan Kellogg on July 6, 2007 - 10:14 am

    The barber lives somewhere else.
    The barber lost all his hair at the age of 13.
    The barber uses a depilatory.
    The barber is an American Indian.
    The barber’s face is a mass of scar tissue.
    The barber is the man who shaves the man who shaves himself.

  7. #7 by MartinC on July 6, 2007 - 10:15 am

    Private Ryan ?

  8. #8 by Saint Gasoline on July 6, 2007 - 11:46 am

    Ah, hell! And here I was thinking I had invented an original paradox when I wrote the masturbation paradox, but it looks like Bertrand Russell beat me to it. How was I not aware of this?! I need to reread my philosophy books, I guess.
    Still, though. I think the masturbation paradox is better.
    A girl necessarily has sex only with those who do not masturbate.
    If she has sex with herself, then she can’t have sex with herself because she just masturbated.
    If she doesn’t have sex with herself, then she doesn’t masturbate and is obligated to have sex with herself.
    Russell should have said it like that.

  9. #9 by ~owl~ on July 6, 2007 - 11:53 am

    Let x shaves y be Shave(x,y). So the question becomes E(x)A(y)(Shave(x,y) = ~Shave(y,y)). Since this is presented as being true for every y, it is true for x, or E(x)(Shave(x,x) = ~Shave(x,x)). Which is false. No paradox, simply a falsity. There does not exist such an x.
    On the other hand…
    Let us recognize that shaving oneself is different than shaving another. The first, call it Autoshave(x), is a single place predicate, while shaving someone else, Heteroshave(x,y), is a two-place predicate. It is noted that Heteroshave(x,x) is false, as one cannot Heteroshave oneself (see Dunc’s comment above).
    Under this interpretation, the question becomes E(x)A(y)(Heteroshave(x,y) = ~Autoshave(y)). When we ask about the particular y that is x, we get E(x)(Heteroshave(x,x) = ~Autoshave(x)); as noted above,Heteroshave(x,x) is false, so ~Autoshave(x) is false, so Autoshave(x) is true. The barber Autoshaves.
    I hope this clarifies things :)

  10. #10 by Warren on July 6, 2007 - 12:07 pm

    The problem is not with the barber or the shaving; the problem is with the premise and the language used to describe it. The language has set up what is, in essence, a false dilemma.
    Besides, the barber is a Hassidic Jew. He never shaves.

  11. #11 by Mark Nutter on July 6, 2007 - 12:53 pm

    I found a magic genie who offered to grant me three wishes. First I wished for good looks, then I wished for wealth. For my third wish, I wished that I only had two wishes.
    Did the genie grant my third wish?

  12. #12 by Kevin Beck on July 6, 2007 - 12:54 pm

    “I was thinking I had invented an original paradox when I wrote the masturbation paradox, but it looks like Bertrand Russell beat me to it…”
    Ha ha! That’s right, he spanked you good! You haven’t invented jack! Bet you feel like a real jerk now!
    Inlight of recent events, I think I should have phrased the paradox a little differently myself:
    “A non-licensed bikini waxer shaves all the whatzits in a village whose owners do not shave their own whatzits.
    So who shaves the waxer’s whatzit?”
    Et cetera.

  13. #13 by blf on July 6, 2007 - 1:05 pm

    So who shaves the waxer’s whatzit?

    The barber.

  14. #14 by Toby on July 6, 2007 - 6:20 pm

    There is a story behind this paradox.
    Gottlob Frege is considered as the inventor of modern logic. On the very point of publishing one of his books, he received in the post a version of the “Barber Paradox” from a young Bertrand Russell. Frege had to insert an erratum in his book, pointing out that some sections were invalid.
    The paradox is often stated in term of sets that are members of themselves. Consider library catalogues in the days when such catalogues were large books.
    Being book repositories, some libraries had catalogues which included themselves in the list of library books.
    Now, suppose you did a catalogues of all library catalogues which did not list themselves.
    Should this catalogues list itself?
    W. Quine pointed out a resolution of the Barber Paradox. Since you deduce a logical contradiction, you could conclude that such a barber could not exist. However, you cannot get around the library catalogue version that way.

  15. #15 by Anonymous on July 6, 2007 - 7:27 pm

    I was going to say that Russell took all the fun out of the barber by forcing mathematicians to redefine naive set theory.
    But then I read the comments.

    The barber.

    Well now, aren’t you aware that Occam’s razor is dangerous for jokes? Never, ever, use the raz… um, never mind.

  16. #16 by Anonymous on July 6, 2007 - 7:27 pm

    I was going to say that Russell took all the fun out of the barber by forcing mathematicians to redefine naive set theory.
    But then I read the comments.

    The barber.

    Well now, aren’t you aware that Occam’s razor is dangerous for jokes? Never, ever, use the raz… um, never mind.

  17. #17 by Anonymous on July 6, 2007 - 7:27 pm

    I was going to say that Russell took all the fun out of the barber by forcing mathematicians to redefine naive set theory.
    But then I read the comments.

    The barber.

    Well now, aren’t you aware that Occam’s razor is dangerous for jokes? Never, ever, use the raz… um, never mind.

  18. #18 by Alan Kellogg on July 7, 2007 - 6:32 pm

    Where, exactly, does it say that the barber cannot shave those who shave themselves? Do those that shave themselves shave themselves all the time? Is the barber outright forbidden from shaving himself just because he also shaves those who don’t shave themselves? Are all intellectuals and philosophers afflicted with a cognitive dyslexia?

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