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From: alderson@netcom16.netcom.com (Richard M. Alderson III)
Subject: Re: Consistency & incompleteness (WAS: Is PGP truly secure?)
In-Reply-To: Jeffrey Goldberg's message of Tue, 23 Jul 1996 18:37:30 +0100
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In article <Pine.ULT.3.93.960723182543.20926V-100000@xdm011>
Jeffrey Goldberg <cc047@ecs.pc.Cranfield.ac.uk> writes:

>[I have added sci.lang to the news.groups line; probably
>comp.security.pgp.discuss can be removed from follow-ups, but I will leave
>that for someone else to do]

I think they need to see this.

>It depends on what you mean by "use".  I use English, and I suspect that its
>grammer may allow uncountably many sentences.  (At the moment, I don't have an
>argument for this, but I think there must be one).

I have seen several responses of the form "English uses only the letters of the
alphabet and therefore can only have countably many sentences."

That is incorrect reasoning, mistaking representation of language for language.

Mr. Goldberg's intuition concerning the *grammar* of English, or of any other
natural language, is correct.  There are, in the grammars of natural languages,
recursive processes available that make it possible in theory to generate
sentences of infinite length.  For example, a sentence S can be imbedded into
another sentence S', in the the following sequence.

1. John is sick.
2. I know that John is sick.
3. I said that I know that John is sick.
4. I know that I said that I know that John is sick.
5. I said that I know that I said that I know that John is sick.
6. I feel sure that I said that I know that I said that I know that John is
   sick.

and so on.

It is the case that there are memory constraints on the *processing* of long
embedded sentences, with stronger constraints on center-embedded sentences[1]
than on tail-recursive ones, but that is a fact of the processor and not of the
grammar (which claim will embroil me in arguments with some linguists who think
that all of grammar should reflect the constraints of the flesh directly).

*****
[1] Sentences of the form S' are said to exhibit center-embedding:

    S:  The rat bit the cat.
    S': The cat the rat bit died.

It is possible to generate long strings of such, but processibility by humans
quickly bottoms out.
*****

So the fact that English, or any other natural language, can be represented in
ASCII or any other writing system has nothing to do with the generative power
of the grammar of the language.
-- 
Rich Alderson   You know the sort of thing that you can find in any dictionary
                of a strange language, and which so excites the amateur philo-
                logists, itching to derive one tongue from another that they
                know better: a word that is nearly the same in form and meaning
                as the corresponding word in English, or Latin, or Hebrew, or
                what not.
                                                --J. R. R. Tolkien,
alderson@netcom.com                               _The Notion Club Papers_
