From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Organization: ORA Corporation
Date: Thu, 15 Oct 1992 18:50:41 GMT

In article <Bvz218.5B6@usenet.ucs.indiana.edu>
chalmers@bronze.ucs.indiana.edu (David Chalmers) writes:

>Given that we could know what system we are, doesn't that demonstrate
>a contradiction, as above, in that it implies that we could know
>the truth of our own Godel sentence?  No, it doesn't, and that's
>because the knowledge comes from *external* means (such as inspection
>of the brain).  The truth of the Godel sentence G for M is equivalent
>to the truth of "M could not demonstrate G by *internal* means" (i.e.,
>more or less by sitting there and thinking about math under its
>own steam.  That M could come to know the truth of G by external
>means is no contradiction at all.  Any smart enough computational
>system (that was consistent and believed it was consistent) could come
>to do exactly the same thing, if confronted by perceptual evidence
>of its own design (or indeed, if told by an ever-reliable God figure
>that G was true).  There's not the slightest contradiction there.
>(Daryl McCullough will probably jump on me now, but I'm ready for him.)

Hmm. Is this some kind of trap?

Anyway, as I have said before, Dave, what I believe is impossible is
the combination of (1) the knowledge of what your formal system is,
and (2) the absolute knowledge that your very own formal system is
consistent. Whether the knowledge comes from an external or internal
source makes no difference. Ultimately, the argument goes as follows.

Consider the following sentence G:

    David Chalmers will never believe this sentence.

Now, some people will complain that G isn't a proper sentence, since
it is self-referential. However, it is clear enough (to me, at least)
what the sentence means, and so I think it is a perfectly good
sentence. (If you don't like the self-referential nature, I can
replace it by "David Chalmers will never believe the sentence
occurring in the email message from Daryl McCullough to David Chalmers
on October 15, 1992." Then in an email message, I will send David
Chalmers the message "David Chalmers will never believe the sentence
occurring in the email message from Daryl McCullough to David Chalmers
on October 15, 1992.")

Suppose that G is false. Then since G is true if and only if David
Chalmers will never believe G, if G is false then David Chalmers will
at some time believe G. Therefore, if G is false, then David Chalmers
must come to believe something false. A bit of purely logical
manipulation gives us:

    If David Chalmers never believes anything false, then G is true,
    and David Chalmers does not believe G.

Now suppose that David Chalmers comes to believe that his beliefs are
sound---that is, he comes to believe that his beliefs are all true.
Then David Chalmers can derive the above facts about G as well as
anyone else. So since he believes A, ("David Chalmers never believes
anything false") and he believes that A implies B ("G is true"), then
by modus-ponens he believes B. Therefore, David Chalmers believes G.
Since G is equivalent to the statement "David Chalmers does *not*
believe G", then the fact that David Chalmers believes G implies that
G is false. Therefore, at least one of David Chalmers' beliefs is
false.

The upshot of this is that if David Chalmers believes himself to be
sound, then he is *not* sound. It doesn't make one whit of difference
whether his belief that he is sound is an internally derived fact, or
whether the belief comes from God.

Daryl McCullough
ORA Corp.
Ithaca, NY

P.S. There is a more formal version of this argument, but ultimately
the argument has the same structure as above. The only tricky part
formally is to get a sentence G with the property that G is true if
and only if David Chalmers does not believe G.
Newsgroups: comp.ai.philosophy