Who Shaves Harry the Barber?

Why are the following two statements inconsistent with each other?

1.     Harry the barber shaves only those who do not shave themselves.

2.     Harry the barber shaves all of those who do not shave themselves.

The key question is, "Who shaves Harry?" On (1), either Harry does not shave at all or he is shaved by someone other than himself. On (2), Harry shaves himself. Thus, (1) and (2) are inconsistent with each other. They cannot both be true (although one or the other could be true).

(1) If Harry shaves only those who do not shave themselves, he does not shave any who shave themselves. So he does not (cannot) shave himself. However, there may be those who do not shave themselves and who Harry does not shave – i.e., they do not shave at all.

There are those who shave themselves (S) and those who don’t (~S). No members of (S) are shaved by Harry because Harry shaves only those who do not shave themselves. (~S) contains three subsets: (A) those shaved by Harry; (B) those shaved by someone other than Harry (since Harry does not shave all of those who do not shave themselves); and (C) and those who are not shaved at all (neither by Harry nor by themselves). Harry must be a member of either (B) or (C) – i.e., either he is shaved by someone other than himself or he does not shave at all. That Harry shaves only those who do not shave themselves leaves open the possibility that there are those who do not shave themselves and who are not shaved by Harry (including those who do not shave at all).

(2) If Harry shaves all of those who do not shave themselves, then it is possible that he also shaves some or all of those who do shave themselves. Thus, it is necessary that Harry shaves himself because if he doesn’t shave himself then he would shave himself (because he shaves all of those who do not shave themselves), which is contradictory. He must be shaved by himself – i.e., he is one of those shaved by themselves, but not one of those shaved by themselves and not by Harry. He must be one of those (the only one) shaved by themselves and by Harry.

Again, there are those who shave themselves (S) and those who don’t (~S). All members of (~S) are shaved by Harry because Harry shaves all of those who do not shave themselves. This rules out the possibility that Harry does not shave at all (because if he did not shave at all, then he would not shave himself, which would mean that he both does and does not shave himself, which is contradictory). In this case, (S) contains two subsets: (A) those shaved by themselves and not by Harry; and (B) those shaved by themselves and also by Harry. Harry must be a member (the only member) of (B) – i.e., he must shave himself.

What if (3) Harry the Barber shaves all and only those who do not shave themselves? What follows from that? (click here)

PowerPoint Presentation on the Barber Puzzle