LINGUIST List 3.729

Mon 28 Sep 1992

Disc: Barber Paradox

Editor for this issue: <>

Directory

1. J.T. de Jong, barber
2. Alex Monaghan, Re: 3.717 The Barber Paradox (Part 2)
3. , RE: More of the barber of Seville
4. Lars Ahrenberg, Re: 3.716 The Barber Paradox

What a fuss about barbers Spanish towns! It does not seem to be a problem to me
especially because in SPANISH there are two verbs: afeitar (to shave someone)
and afeitarse (to shave oneself). In this so-called paradox there are simply
two different verbs: S[i]VO[j] and S[i]VO[i]. With two verbs there is no
paradox.
 _________________________________________________________
Jelly Julia de Jong, Dept. of General Linguistics, University of Groningen
Oude Kijk in 't Jatstraat 26 E-mail: julialet.rug.nl
9712 EK Groningen The Netherlands

Message 2: Re: 3.717 The Barber Paradox (Part 2)

i must confess to being a little surprised at the fuss about this one.

imagine a mediaeval sevilla where anti-semitic paranoia has reached
such a peak that beards (the mark of a jew or arab) are to be outlawed.
the church (or some other suitable tyrant) issues the following edict:

	EVERY MAN WHO DOES NOT SHAVE HIMSELF WILL BE SHAVED BY
		THE BARBER-GENERAL OF THE INQUISITION.

	(presumably with red-hot needles or the like ...)

as far as i can see, this gets rid of all the beards but has no other
necessary consequences like over-working the poor barber-general or
forcing him to buy a mirror. ... any questions?

								alex.

p.s. bet you didn't expect the spanish inquisition!

> 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

These days there is an increasingly popular way out of this paradox: the
fuzzy logic way. Instead of admitting only TRUE or FALSE (binary 0 or 1),
admit that there are degrees of truth (multivalency [0,1] : any numbers
between 0 and 1). Thus,

Let S be the proposition that the barber shaves himself and not-S that he
does not. Then since S implies not-S, and not-S implies S, the two
propositions are logically equivalent: S = not-S. Equivalent propositions
have the same truth values:

 t(S) = t(not-S)
 = 1 - t(S)

Solving for t(S) gives: t(S) = 1/2 !

References:

1) Klir, G.J & Folger,T. A. "Fuzzy Sets, Uncertainty, and Information" 1988.
2) Bart Kosko, "Neural Networks and Fuzzy Systems--A Dynamical Systems
 Approach to Machine Intelligence" 1992.

Incidentally, I see an increasing role for fuzzy logic in Natural Language
Processing systems where it can handle not only paradoxes but also the more
mundane things like linguistic hedges ("very", "many", etc.)

Cheers,
An Nguyen
Lex-Kon, Inc.
e-mail: anverbum.com

Message 4: Re: 3.716 The Barber Paradox

Steve Berman in his enlightening reply Guy's original posting about the
Barber Paradox ends by saying

> (Of course, there's no contradiction if the barber is removed from the
> domain of quantification -- 'every man except for the barber')

Now, I'd like to suggest that this is the way we normally, or at least,
often, interprets statements with a universal quantified NP and an individual
NP in two different argument positions. Compare

 John watched everybody in the room, or
 No candidate is as good as Brown.

In the first case John may well be in the room, but the quantified NP would
be understood as referring to the other people in the room, and in the second
case, assuming that Brown is a candidate, the quantified NP would be understood
as applying to all the other candidates. It would be quite pointless
to rebut by saying things like "Oh, he couldn't as there was no mirror in the
room" or "But surely Brown is as good as Brown" (unless, of course, you would
like to show off as someone knowing logic).

A reasonable response to the question "Does the barber shave himself?" or,
alternatively, "Who shaves the barber?" is thus that the premise does not
give us enough information. One can argue, though, that the fact that the
question is asked implies that it should do, and therefore forces us to
look for interpretations of the premise that apply to the barber as well.

-- Lars Ahrenberg