There is a club called the Exclusive Club. Somebody is a member of this club if and only if he has not shaved anybody who has shaved him. In other words, X is a member of the Exclusive Club if and only if there is no Y such that X shaves Y and Y shaves X.
A barber once claimed that he had shaved every member of the Exclusive Club and nobody else. Show that the barber's claim cannot be true.
(In reply to re(4): solution
I accept your interpretation , but consider it "weak".
Nothing in the text IMHO precludes the following understanding:
Club members belong to a subset of people not shaving people who did not shave them(PNSPWDNT). This subser we call EXI.CLUB.
There might exist some PNSPWDNT out of this subset.
Using a famous paradox as a basis of logic-oriented puzzle is quite questionable.