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- Tom Lai, Re: 3.711 Queries: Paradoxes
- "David M. W. Powers", The barber is a woman!

> From: CPCCVX::CTTOMLAI 22-SEP-1992 10:19 > > For every individual x in the Spanish town, > If x does not shave x > Then Pablo (the barber) shaves x > I must add that the paradox requires further that If Pablo (the barber) shaves x Then x does not shave x This does not follow from the way in which the problem was described (in words) by Jacques Guy. But the paradox as appears in the literature should require the conditional to be bi-directional. I can try and find a more accurate (verbal) rendering of the paradox if there's a need to do so. Tom Lai, City Polytechnic of Hong KongMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

>A long time ago, somewhere (I thought it was in Hofstadter's "Goedel, >Escher, Bach", but I can't find it there), some years ago then, in some >book riddled with mathematical brain-teasers, I read about the Spanish >barber. The story goes like this: > > There is a small Spanish town where every man who does not shave > himself is shaved by the barber. Does the barber shave himself? No! Because the barber cannot be a man (and I make the charitable assumption that she doesn't need to shave). So the linguistic trick is that the neutral reflexive in English implies the male reading. Are you sure, Jacques, that you didn't encounter it in Bob Kowalski's 1979 book "The Logic of Programming"? There a similar example is used to illustrate that there are problems which (i) can't be expressed in Horn Clauses (Definite Clause Logic) effectively, and (ii) can't be solved by resolution alone (without merging, factoring or equivalent). It was expressed slightly differently (from memory): Everyone either shaves himself or is shaved by every barber. Nobody both shaves himself and is shaved by a barber. (Viz. above should be read as exclusive or.) Show that there can be no barber. (Or equivalently, how many barbers must there be?) Using proof by contradiction, we express this in clausal (PROLOG type) notation (where ":- is read as "if", "," is read as "or" on the left and "and" on the right - actually "," is always "or", but the terms to the right of the ":-" are negated - it is a conjunctive normal form, in which all variables are exclusively quantified within a clause): s(B,X), s(X,X) :- b(B). If B is a barber, B shaves X or X shaves X. :- s(B,X), s(X,X), b(B). If B is a barber, not B shaves X or not X shaves X. b(bob). bob is a barber (the claim we want to contradict). resolving on the b(B) terms gives: s(bob,X), s(X,X). :- s(bob,X), s(X,X). Factoring both (unifying terms of a clause to give a special case or factor): s(bob,bob). :- s(bob,bob). Resolution leads to a contradiction. QED. (N.B. In this notation the convenient PROLOG "if" reading is a little awkward, and you need to know that an empty left hand side means FALSE, and an empty right hand side means TRUE - this follows if you remember that a clause is a disjunction ("or" of terms, AT LEAST ONE of which must be true) and the left hand context retains this character whilst the right hand context becomes conjunctive ("and" of terms, NOT ONE of which may be false). I don't really think the "paradox" has much to do with language, and the problem can be expressed in many different ways. However our use of quantifiers can be a problem. There is a tendency to interpret, at first blush, a (partial) formulation like "All the barbers shave every man who does not shave himself." as if it were a unionist collectivist statement that the barbers of Seville (or wherever) have a monopoly. My favorite antimony is the story of the hanged prisoner, and one of the best discussion I've seen of antimony and paradox is Quine's "The Way of Paradox" (if I remember aright) - although I'm not happy with his explanation of the prisoner's dilemma: A judge sentenced a prisoner to be hanged at dawn one day in the coming week, with the stipulation that he would not know on what day it would be until he was fetched for execution, but it would be an unpleasant surprise. He reasoned that it couldn't be the Saturday, as otherwise he'd know by sunrise on the Friday. Thus it must be one of the first six days of the weeks. By successive reasoning backwards through the week he eliminated each day in turn, concluding that the judge's sentence could not be carried out. He was thus most surprised when he was executed on the Tuesday. Have fun! dP P.S. Quine's explanation is that the prisoner ASSUMED implicitly that he could work out what day he would be executed, viz. the falsity of the hypothesis, and thus his deduction of unsatisfiability of the sentence is not a contradiction itself, but a consequence of FALSE or CONTRADICTORY assumptions. I think that was Quine's answer, it is ten years or so ago since I read about it and I've seen various attempts at explanation, none of which I have found completely satisfying. In other words, that's why it's my favourite paradox. -- Dr David M. W. Powers +49-631-13786 (GMT+1) E xtraction Auf der Vogelweide 1 +49-631-205-3210 (FAX) O f SHOE W-6750 KAISERSLAUTERN FRG powersdfki.uni-kl.de H ierarchical S tructure for Machine Learning of Natural Language and OntologyMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue