LINGUIST List 2.728

Wed 30 Oct 1991

Disc: Infinite Languages

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1. , Infinite Language
2. Henry Kucera, Re: 2.714 Is Language Finite? Motion Verbs
3. , Infinite languages

Michael Kac, as usual, hits the nail on the head (in fact, more than
one: there is going to be an epidemic of Kopfschmerz in the Nagel
family) in his recent posting. The rules of football do not impose
an upper bound of permissible scores (perhaps we should stop using
the word 'possible' in the sense of 'legal' or 'permissible'),
and a program for adding two integers might well be written in a way
that did not involve imposing an upper bound on the size of the
integers. In fact, the computer analogy is the relevant one,
because some (many) programs do impose such bounds, whereas others
do not. For example, the DOS operating system can only address
so much memory and can only handle calendar dates up to a fixed
point (sometime in the next century, I forget the details).
 Thus, contrary to some people in linguistics have asserted over
the years, the computer analogy does NOT support the standard
Chomsky position on unbounded length of sentences. It merely shows
that this position is not inherently untenable. But it also shows
that the opposite position is also tenable.
 Now, I am not at all sure that this issue can be resolved on
factual grounds in all cases (as I have been arguing all along),
but there are certainly special cases where it can be resolved
PROVIDED the construction we are studying DOES have an upper
bound. If it does NOT, then I don't see any way of conclusively
showing that this is the case.
 Also, what complicates the picture is that the linguist's
usual (and I think mistaken) distinction between competence and
performance makes it very difficult to draw ANY conclusions from
any given piece of data.

Message 2: Re: 2.714 Is Language Finite? Motion Verbs

>Date: Wed, 23 Oct 91 14:54:42 -0500
>From: "Michael Kac" <kaccs.umn.edu>
>Subject: Re: 2.686 Is language infinite
>
>In response to Henry Kucera's query re Hockett's baseball/football analogy:
>Hockett, if I remember correctly, argues that football scores above a certain
>number are impossible because a game of football is played within a closed
>time interval (whereas a baseball game continues until one side wins, howe-
>ver long it takes). The problem I have always had with this argument is that
>I see an equivocation in 'impossible'. It's certainly true that for team of
>human football players to score, say, 1m. points in a single game would be
>impossible from a performance standpoint. But insofar as the rules of football
>are concerned, such a score is perfectly possible.
>
 This is part of Michael Kac's argument. I think that it, in effect, goes to
the heart of the matter: As far as I know, only humans play football and,
similarly, it is humans who speak natural languages. This is not, in my view,
a point to be dismissed lightly. We are, after all, dealing with a specific
biological and mental facility particular to humans. Is it not the case that,
in the most elementary sense, the limits of human articulatory organs restrict
the set of possible sounds that a language might have? Or that it explains
certain historical developments, such as palatalizations? Or, on the mental
level, that human memory limitations pretty much account for *all* languages
being highly redundant in terms of information theory (more than 80%
probably)? Or even: if a universal "faculty of language" actually exists, is
it not bound to humans and thus essential to an understanding of language
acquisition, for example (if one believes in a universal grammar theory)?
 And that, consequently, some languages (such as those in which every well-
formed sentence must have an even number of words) do not exist? (One could
very easily design a programming language with exactly this limitation--as a
matter of fact, all "machine-level" languages have precisely some such
property.
 I can't help but feel that the discussion on this issue has not been much
different from medieval theology. I brought up the Hockett argument because
I am increasingly concerned about the reaction of the "real world" out there to
our linguistic abstractions. Is it any wonder that many psychologists or
 even computer scientists are not taking us very seriously and that deans want
to derminate our departments?
 Henry Kucera

There was a recent request for an evaluation of Hockett's claims
about finite and infinite languages in his book State of the Art.
While Michael Kac has commented on one aspect of this, I would
like to add, having located and reread the book in question, that
Hockett's main point seems to be that the notions of finiteness
and infinity (and indeed all of the theory of computation) cannot
be applied to sets that are not well-defined. At the same time,
he claims that languages are not well-defined (and incidentally
that no physical system is well-defined). At the same time<
Hockett argues that languages are finite, specifically, that
very long sentences are not grammatical (w/o, of course, a specific
upper bound on their length).
It seems to me that, while much that he says makes sense, Hockett
cannot have it both ways. He cannot assert that languages are finite,
and also that the finite/infinite distinction does not apply to them.
Furthermore, I think that he is utterly wrong in his understanding
of the relevant mathematical notions. That is, there is nothing
to prevent us from talking about infinite sets that are not entirely
well-defined. The easiest way to do this is to refer two well-defined
sets A and B, both of which are infinite and such that A is a proper
subset of B, and then refer to a third set C, for which we have
no precise definition, but of which we know that it is a superset
of A and a proper superset of B, for example. Then C is inescapably
infinite.
This is much like my earlier argument that there are sets without
definite boundaries which are clearly finite because we know that
they are contained within well-defined finite sets. But the sword
here cuts both ways.