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I'm reading To Mock a Mockingbird, by Raymond Smullyan, and I'm puzzled by the solution to one of the problems.

Chapter 3, Problem 5, The Exclusive Club, states:

There is [a] club known as the Exclusive Club. A person is a member of this club if and only if he doesn't shave anyone who shaves him. A certain barber named Cardano once boasted he had shaved every member of the Exclusive Club and no one else. Prove that his boast involves a logical impossibility.

The author argues that this is a logical impossibility because, if Cardano's statement is true, it leads to a contradiction. Firstly, he would not be a member of the Exclusive Club because he would have shaved himself (having shaved all members of the Club), which no member of the club can do. Neither has Cardano ever shaved himself.

Assuming his statement is still true, the author argues the contradiction occurs because someone from the Exclusive Club will then have shaved Cardano. This person cannot therefore be a member of the Exclusive Club, having shaved someone (Cardano) who has shaved everyone in the Club. Thus Cardano's claim is contradictory and false.

I fail to see how this argument holds up. I see at least two possible, logically valid, scenarios where Cardano's claim is true and not contradictory.

  1. Cardano is not a member of the Exclusive Club. Cardano has shaved everyone in the Exclusive Club, but no one in the Exclusive Club has shaved him. Someone who is not in the Exclusive Club has shaved Cardano and Cardano has shaved that person as well.
  2. No one has ever shaved Cardano.

Am I missing something?

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Your first possibility doesn't work cause at the end you say someone who is not in the exclusive club has shaved Cardano and he shaved them back. This violates his claim that he has shaved everyone in the exclusive club and no one else.

Your second doesn't work since if no one has ever shaved Cardano, then he hasn't shaved anyone who has shaved him (since no one has shaved him). Thus by definition he is in the exclusive club. But this violates Smullyan's (I think valid) conclusion that he is not in the exclusive club: for if he were, by his claim, he must have shaved himself and thus violated the condition for membership in the exclusive club.

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