LINGUIST List 3.716

Wed 23 Sep 1992

Disc: The Barber Paradox (Part 1)

Editor for this issue: <>

Directory

1. Stavros Macrakis, 3.711 Queries: English Dialect Syntax, Paradoxes, Norman fix
2. Steve Berman, 3.711 Queries: English Dialect Syntax, Paradoxes, Norman fix
3. Dale Russell, 3.711 The barber's paradox
4. William Dowling, The Barber (paradox regained)
5. , Re: 3.711 Queries: English Dialect Syntax, Paradoxes, Norman fix

J.Guy asks about the barber's paradox: "Or is it me who has gone soft
in the brain?" I doubt that's the problem, but his analysis is
incorrect. There is no "sleight of words in the use of the reflexive
pronoun". Let's forget about towns and barbers and shaving and look
at the problem abstractly.

	Premise: NOT f(x,x) IFF f(x,B).

	Now, is f(B,B) true?

	Well, let's plug in B for x in the premise:

	 NOT f(B,B) IFF f(B,B).

	This is a contradiction, so the original definition of "f" is
	flawed.

 -s

PS By the way, the "mathematical brain-teaser" (in set form, rather
than predicative form) is actually Bertrand Russell's demonstration
that Cantor's set theory had a fatal flaw, a major event in the
history of logic.

PPS _Goedel, Escher, Bach_ is a bad book. It takes such simple but
deep ideas as the barber's paradox and Goedel's theorem and universal
Turing machines, and dresses them up with empty rhetoric and
mystification ("whispering in awe of the mysteries of set theory", as
J. Guy puts it) so you miss the basic point.

Jacques Guy gives the paradox of the barber as follows:

	 There is a small Spanish town where every man who does not shave
	 himself is shaved by the barber. Does the barber shave himself?

and this explanation of it:

	 Either the barber shaves himself or he doesn't. If he does not
	 shave himself, then he is shaved by the barber; therefore he shaves
	 himself. But if he shaves himself, he is not shaved by the barber,
	 so...

He suggests, however, that there is no paradox:

 The argument makes sense if, and only if,
 you subscribe to the hidden assumption that the barber is not himself.
 That is nothing but a sleight of words in the use of the reflexive pronoun.
 Avoid the use of pronouns and the paradox disappears. Let every man in
 the village have a distinctive name, and let's call the barber Pablo.

 For any man there can be only two cases:

 1. He shaves himself, viz Pepe shaves Pepe.
 2. Pablo shaves him, viz Pablo shaves Pepe.

 What about Pablo? Those two cases are one and the same:

 1. He shaves himself, i.e. Pablo shaves Pablo.
 2. Pablo shaves him, i.e. Pablo shaves Pablo.

 Isn't that what we may call a linguistic illusion, or sleight of
 hand?

Like Guy, I have been acquainted with this story for some time but am also
unable to find it cited in any of the literature I have at hand; the
following remarks are just my spontaneous response to Guy's representation
of it.

It is easy to see what is going on if we translate the story into a
first-order predicate calculus notation ('A' is the universal quantifier,
'S' is the binary relation of shaving, 'b' is the barber). The version
related by Guy may be rendered as:

 Ax(-Sxx --> Sbx)

If we add the premise that the barber does not shave himself (-Sbb) we get
an immediate contradiction by universal instantiation and modus ponens. But
no contradiction arises if we add the premise that the barber shaves
himself. Therefore, the version given above is no paradox. However, the
following version is paradoxical (i.e., contradictory), as its predicate
calculus translation coupled with either of the premises -Sbb or Sbb shows:

 Every man who doesn't shave himself is shaved by the barber but every
 man who does shave himself is not shaved by the barber.
 Ax((-Sxx --> Sbx) & (Sxx --> -Sbx))

This version of the paradox is implicit in Guy's explanation; however, in
his attempt to demonstrate that there's no paradox, he reverts to his
original statement, i.e., leaving out the crucial second conjunct. This
version (i.e., with both conjuncts) is also equivalent to

 Ax(Sxx <--> -Sbx)

i.e., 'every man shaves himself if and only if the barber doesn't shave
him', which makes the paradoxicality even plainer. As for the version given
by Guy, we might allow that it is also paradoxical by 'linguistic sleight of
hand', that is, by a kind of implicature of the natural language idiom that
makes the original (material) implication understood as the contradictory
biconditional. (Of course, there's no contradiction if the barber is
removed from the domain of quantification--'every man except for the barber'.)

--Steve Berman

There is indeed a hidden, and invalid assumption, at least in the way
the paradox is formulated here.

 >> There is a small Spanish town where every man who does not
 >> shave himself is shaved by the barbar.

On reading this, we ASSUME that the converse is also true; that every
man who does shave himself is not shaved by the barbar. But this is
nowhere stated. The sentence is just silent about men who do shave
themselves. And it turns out that the barber is just such a case. He
is a man who does shave himself, and is also shaved by the barber. So
in this formulation, there is no paradox at all.

Maybe this is just another way of describing what you claimed to be
the hidden assumption, `that the barber is not himself.' I had a hard
time figuring out what that meant. But the example with Pepe and
Pablo looks correct.

But as you noted, this so-called paradox has been put forth many
times, and probably stated more carefully in some formulations than
others. If the statement of the problem had been

 >> There is a small Spanish town where ALL AND ONLY those men who
 >> do not shave themselves are shaved by the barbar.

then I see no way around the paradox. That is, if you accept another
hidden (and invalid) assumption, that the barber is a man. If the
barber is a woman, then, again, the paradox goes away. But of course,
that's just due to insufficient precision in the statement of the
scenario.

 Dale Russell

Re the recent posting of j.guytrl.oz.au (Jacques Guy) on the Barber paradox:
The statement
> There is a small Spanish town where every man who does not shave
> himself is shaved by the barber. Does the barber shave himself?
indeed is not contradictory, since it admits the possibility that the barber
may shave other folks in addition to those who don't shave themselves. In
particular, this allows the loophole that he shaves himself, as was pointed
out. Stated a little more carefully, however, there really is a
contradiction:
	There is a town in which there is a barber who shaves all and only
	those who do not shave themselves.

This is how I have explained this paradox (and the heterological paradox, and
the halting problem, and in fact all diagonalization arguments) to students:

You have a square matrix M, all of whose cells contain either a 1 or a 0. Let
M[i,j] (that is the cell in column j of row i) have a 1 if resident i (of our
small hypothetical Spanish town) shaves resident j. Thus if there were a
peculiar fellow (P) who (quite the opposite from the putative barber) shaved all
and only those folks who shaved themselves, there would be a row (P) of the
matrix, which, cell-for-cell, was equal to the diagonal: M[P,1] = M[1,1];
M[P,2] = M[2,2]; ... etc. The existence of such a guy, however odd, is not
contradictory. But on the other hand, if there were a barber Q who shaved all
and only those people who did _not_ shave themselves, then we would be saying
for some row Q of the matrix, M[Q,1] ~= M[1,1] and M[Q,2] ~= M[2,2], etc. But
the diagonal of any matrix intersects all its rows: looking at row Q, column
Q, we have M[Q,Q] ~= M[Q,Q]. That is the contradiction. So there can be no
such guy Q. Another way of saying it is that no row of a binary matrix can
be the negation of its diagonal.

If you have a taste for such things, you can prove Kleene's recursion theorem
this way, too.

Will Dowling (willfranklin.com)

Message 5: Re: 3.711 Queries: English Dialect Syntax, Paradoxes, Norman fix

> From: j.guytrl.oz.au (Jacques Guy)
> Subject: The barber's paradox: a linguistic illusion?
>
> There is a small Spanish town where every man who does not shave
> himself is shaved by the barber. Does the barber shave himself?
>

> ... and so on ad nauseam. The conclusion reached, after invoking the manes
> of Bertrand Russell and whispering in awe of the mysteries of set theory,
> was that there can exist no such town. The Spanish barber paradox has
> always seemed vacuous to me. The argument makes sense if, and only if,
> you subscribe to the hidden assumption that the barber is not himself.
> That is nothing but a sleight of words in the use of the reflexive pronoun.
> Avoid the use of pronouns and the paradox disappears. Let every man in
> the village have a distinctive name, and let's call the barber Pablo.
>
> For any man there can be only two cases:
>
> 1. He shaves himself, viz Pepe shaves Pepe.
> 2. Pablo shaves him, viz Pablo shaves Pepe.
>
> What about Pablo? Those two cases are one and the same:
>
> 1. He shaves himself, i.e. Pablo shaves Pablo.
> 2. Pablo shaves him, i.e. Pablo shaves Pablo.
>
> Isn't that what we may call a linguistic illusion, or sleight of
> hand?

I don't think that is really a linguistic illusion.

I owe Jacgues Guy a favour for using his COGNATE, but I cannot agree with
his view here.

As presented in J. Guy's posting, the 'paradox' is reduced to:

	For every individual x in the Spanish town,
	Either x shaves x
	Or Pablo (the barber) shaves x

And hence the 'paradox' is not a paradox.

The problem is that the above interpretation of the paradox is not
correct. The correct (sorry for having to use this word, but _ich kann
nicht anders_) interpretation is:

	For every individual x in the Spanish town,
	If x does not shave x
	Then Pablo (the barber) shaves x

The paradox is that

	Pablo shaves Pablo

and

	Pablo does not shave Pablo

both lead to contradiction.

Tom Lai, City Polytechnic of Hong Kong