LINGUIST List 3.717

Wed 23 Sep 1992

Disc: The Barber Paradox (Part 2)

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

Message 1: Re: 3.711 Queries: Paradoxes

Date: Tue, 22 Sep 92 13:45 +8
From: Tom Lai <CTTOMLAICPHKVX.bitnet>
Subject: Re: 3.711 Queries: Paradoxes

> 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 Kong
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Message 2: The barber is a woman!

Date: Tue, 22 Sep 92 00:27:09 METhe barber is a woman!
From: "David M. W. Powers" <dp%laptoprusvm1.rus.uni-stuttgart.de>
Subject: The barber is a woman!

>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 Ontology
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