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From: goldfain@osiris.cso.uiuc.edu
Newsgroups: comp.ai
Subject: Re: Langendoen and Postal (posted by: B
Message-ID: <8300011@osiris.cso.uiuc.edu>
Date: Wed, 4-Nov-87 01:31:00 EST
Article-I.D.: osiris.8300011
Posted: Wed Nov  4 01:31:00 1987
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Nf-From: osiris.cso.uiuc.edu!goldfain    Nov  4 00:31:00 1987


> /* Written 10:34 am  Nov  1, 1987 by berke@CS.UCLA.EDU in comp.ai */
> /* ---------- "Langendoen and Postal (posted by: B" ---------- */
> I just read this fabulous book over the weekend, called "The Vastness
> of Natural Languages," by D. Terence Langendoen and Paul M. Postal.
>    ...
> Their basic proof/conclusion holds that natural languages, as linguistics
> construes them (as products of grammars), are what they call
> mega-collections, Quine calls proper classes, and some people hold cannot
> exist.  That is, they maintain that (1) Sentences cannot be excluded from
> being of any, even transfinite size, by the laws of a grammar, and (2)
> Collections of these sentences are bigger than even the continuum.  They are
> the size of the collection of all sets: too big to be sets.
>            ...
> /* End of text from osiris.cso.uiuc.edu:comp.ai */

Hang on a minute!   It *sounds* as though  you are talking  about Context-Free
Grammars/Languages (CFGs/CFLs) here.  Most linguists  (I'd wager) set up their
CFGs as admitting  only finite derivations  over a  finite  set of  production
rules, each rule only allowing finite expansion.  Thus, although usually a CFL
is only a  proper subset of  this, we are   ALWAYS working  WITHIN the  set of
finite strings (of  arbitrary  length) over a  finite alphabet.
     Such a set is countably infinite.  Far from being a proper class, this is
a very  manageable set.  If you  move the discussion  up to the cardinality of
the  set of "discourses", which would   be finite  sequences of strings in the
language, you are still only  up to  the power set of the  integers, which has
the same cardinality  as the  set of Real numbers.  Again,  this is a set, and
not a proper class.
     I haven't seen the book you cite.  They must make some argument as to why
they  think natural  languages  (or linguistic   theories about  them)   admit
infinite sentences.  Even given that, we  would have only  the Reals (i.e. the
"Continuum") as a cardinality without some further surprising claims.  Can you
summarize their argument (if it exists) ?

           Mark Goldfain                    arpa: goldfain@osiris.cso.uiuc.edu
           Department of Computer Science
           University of Illinois at Shampoo-Banana