For reference: The Barber Paradox.

The Barber is he who shaves all those, and only those, who doesn't shave themselves.

Now the question is: Who shaves the Barber?

The paradox being that if the Barber shaves himself he must be one of those who doesn't shave themselves, and if he does shave himself he cannot be one of those who the Barber shaves.

So my question is: Is this really a paradox or simply a case of category error?

Because a barber shaves people for payment, yet he would not pay himself when he shaves himself, so therefore when shaving himself he is not de facto "The Barber". The Barber is in fact primarily a person and secondarily (sometimes) The Barber.

  • 2
    The paradox mentions nothing about payment. Assume he works for free, there are also multiple ways to rephrase it so that such analogical distractions do not come up. Indeed, it was Russell's way of popularizing a purely mathematical paradox, see Wikipedia. – Conifold Aug 7 '18 at 6:00
  • I argue that "sets that do/don't include themselves" does not exactly fit what's going on here. The paradox as stated is framed in natural language and as such are predicated on natural language definitions. And then a barber that does nothing but shave, and does so for free, or a shaving machine that also grows a beard, all seems very contrived... I understand there is a real problem with Cantor Set Theory, but Russel's solution to that is actually quite similar to the solution above. But one arise from improper handling of sets and this paradox depends on the conflagration of categories – christo183 Aug 7 '18 at 6:54
  • The barber is just a popular illustration, everything that natural language associations add to it is irrelevant. That cutting them off makes it contrived is fine as long as it can still serve the illustrative purpose. If natural associations suggest a solution that is nice, but solution is a separate issue from the statement. And Russell's solution did not involve introducing extraneous distinctions analogous to payment, that would not work for reformulations where payment makes no sense. It involved the vicious circle principle. – Conifold Aug 7 '18 at 7:16
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    He's shaved by Occam's razor. – user4894 Aug 7 '18 at 18:20
  • 1
    Presumably the Barber doesn't shave. Either that or the situation is impossible. I find it a poor scenario for exploring R's set-theoretic paradox since it doesn't seem to be a paradox. – PeterJ Aug 14 '18 at 13:50

You don't even need to use the word "barber," or to assume any exchange of money in this case. It takes a bit of elementary set theory (logic alone doesn't seem to be enough), but it is easy to prove that someone living in the village can shave those an only those men living in a village who do not shave themselves if and only if that person is not a man [edit: e.g. that person could be a woman living there]. (Full details at my blog posting.)

  • Or the barber could be a machine. For some other caveats see here: en.wikipedia.org/wiki/Barber_paradox, And the exchanges between me and Conifold which also tells the real purpose of the Barber paradox: Illustrating contradictions in Cantor Set Theory. – christo183 Aug 14 '18 at 6:02
  • @christo183 That would work if a machine could be a villager, maybe some kind of android? The biconditional in myproof only works for elements of the set of villagers V, who may or may not be men, i.e. elements of the subset M. – Dan Christensen Aug 14 '18 at 12:44

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