LINGUIST List 3.750

Wed 07 Oct 1992

Disc: The Barber Paradox

Editor for this issue: <>

Directory

1. , Barber Paradox
2. Tom Lai, Barber's Paradox
3. Stephen P Spackman, Re: 3.741 The Barber Paradox: Sign and Referent

There has been some discussion on the list recently of the sources of
the `Barber Paradox'. This is discussed by Quine, who writes (in `The
Ways of Paradox' [first published as `Paradox' in _Scientific
American_ (Volume 206, 1962)], _The Ways of Paradox and other essays_,
Random House, New York, 1966, page 4):

		As a first step onto this dangerous ground, let
	us consider another paradox: that of the village barber.
	This is not Russell's great paradox of 1901, to which we
	will come, but a lesser one that Russell attributed to
	an unnamed source in 1918. In a certain village there
	is a man, so the paradox runs, who is a barber; this barber
	shaves all and only those men in the village who do not
	shave themselves. Query: Does the barber shave himself?

Russell's original discussion can be found in `The Philosophy of
Logical Atomism' (originally presented as a lecture series in
London in early 1918, then published in _The Monist_ (volumes 28 and 29)
in 1918 and 1919, and reprinted in _Logic and Knowledge: Essays 1901-1950_,
edited by Robert C. Marsh, Capricorn Books, New York, 1956). On page
261 of the Marsh collection, speaking of `the contradiction about classes
that are not members of themselves', Russell writes:

		That contradiction is extremely interesting. You can modify
	its form; some forms of modification are valid and some are not.
	I once had a form suggested to me which was not valid, namely the
	question whether the barber shaves himself or not. You can define
	the barber as `one who shaves all those, and those only, who do not
	shave themselves'. The question is, does the barber shave himself?
	In this form the contradiction is not very difficult to solve....

One can find more extensive discussions of paradoxes of mathematical
and philosophical logic in Russell's 1908 paper `Mathematical logic as
based on the theory of types' (reprinted in the Marsh collection and
in J. van Heijenoort, ed., _From Frege to G\"{o}del: A Sourcebook in
Mathematical Logic, 1879-1931_, Harvard University Press, Cambridge,
Mass., 1971), in S.C. Kleene, _Introduction to Metamathematics_), and
in recent writings of Raymond Smullyan, such as ... what was the name
of that book?

Wishing I could find a closing salutation,
Dick Oehrle

Excuse me for giving the following largely non-linguistic account
of the paradox of the barber of Seville.

This paradox as studied in Mathematics is a result of an inherent
problem with defining a set as

 {x|x possesses certain properties}

where x is an individual in the universe of entities under study (and
"|" means "such that").

This way of defining a set gets into trouble when we define the set S as

 S = {x|x is not a member of x}

i.e. S is the set of sets that are not members of themselves (please
interpret this comment in conjunction with the notational definition
given above).

The paradox is that S can neither be a member of itself nor not be a
member of itself. More specifically,

 x is in S <=> x is not in x ("<=>" is just "if and only if")

Subsituting S for x (to find out whether S itself is in S), we have

 S is in S <=> S is not in S (#)

Coming back to the Spanish barber, the set of (all) inhabitants of the
town who are shaved by the barber is

 Q = {y | y does not shave y}

Thus, if the barber is b,

 b shaves y <=> y does not shave y

To find out whether b is in Q or not, we substitute b for y and get

 b shaves b <=> b does not shave b (##)

(##) is the same kind of paradox as (#).

_SO_, there is a real paradox here. How to give an accurate verbal
(linguistic) account of the barber's version of this paradox may not
be trivial. But whether an attempt to do so is successful or not, the
paradox is there for its own sake.

Tom Lai, City Polytechnic of Hong Kong (Sorry for this _long_ posting)

Message 3: Re: 3.741 The Barber Paradox: Sign and Referent

In article 3.741.1 of this list, "the Barber Paradox as confusion
between sign and referent", j.guytrl.oz.au (Jacques Guy) would have us
believe that because the Barber Paradox and its relatives force us to
the conclusion that there is an additional category of things in the
universe than those allowed by the original phrasing of the problem, it
is therefore not a paradox but a linguistic sleight of hand. Perhaps he
fails to grasp that this is precisely the nature of paradox and that
this conclusion is the one to which the mathematical philosophers (and,
indeed, Doug Hofstadter) would have us lead? The precise (classical)
"fix" to the problem of the set of all sets that do not contain
themselves is a heirarchy of set-LIKE entities arranged so that the
paradoxical self-embedding objects do not exist - the addition, that is,
of NEW categories not stated in the problem.

It is one of the most important tools of (classical) mathematics to set
up a system that is "paradoxical" - inconsistent - in order to
demonstrate that a set of premisses cannot ALL be true. It is perhaps
unfortunate that this methodology does not reliably discriminate between
methodological errors, false premisses, and equiplausible but mutually
inconsistent hypotheses (and this may be the psychology underlying the
rejection of this methodology by some mathematicians), but that doesn't
make it either a "sleight" or particularly "linguistic".

Of course, one might adopt the stance of an ultra-Whorfian and suppose
that the structure of arithmetic is in fact determined by some
coincidental feature of the languages of all cultures that have explored
number theory - but there are enough of these and sufficiently varied
that this feature would seem to have the status of an "accidental
universal" - it would require (in light of the evidence) something of an
intellectual leap to suppose that there are languages with the property
that their speakers are LOGICALLY INCAPABLE of verifying the proofs of
the various "anti-recursion" theorems.

(Actually, of course, what is going on is that the human mind is finite
and we can only do APPROXIMATIONS of mathematics. Those who hold to the
doctrine of infinite language are forced to postulate that we can
equally aspire only to approximations of language. Small wonder we get
confused occasionally.)
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stephen p spackman stephenacm.org
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