From newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!uwm.edu!spool.mu.edu!agate!stanford.edu!Csli!avrom Tue Nov 24 10:51:23 EST 1992
Article 7582 of comp.ai.philosophy:
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>From: avrom@Csli.Stanford.EDU (Avrom Faderman)
Subject: Re: Human Intelligence vs. Machine Intelligence
Message-ID: <1992Nov10.224028.18204@Csli.Stanford.EDU>
Organization: Stanford University CSLI
Date: Tue, 10 Nov 1992 22:40:28 GMT
Lines: 43

Sorry not to quote, but I have lost the original articles.  There has been 
a lot of talk in the group about Godel's theorem recently, and, in particular, 
about whether certain sentences qualify as "Godel sentences for the human 
mind."  One in particular is the Quine sentence, "Diagonalizing 'diagonal-
izing X yields a sentence that [the person whose incompleteness we are 
trying to demonstrate] cannot consistently believe' yields a sentence that 
[the person whose incompleteness we are trying to demonstrate] cannot 
consistently believe."

Some people have argued that this sentence does not do the trick.  If, 
they have said, we view the kernel as just a string of words, we must 
view the resultant sentence just as a string of words, and strings of 
words aren't really _believed_ by anybody (rather, the propositions that 
they express are).  On the other hand, if we regard the kernel as having 
a content, the whole sentence is about its own content, and is therefore 
meaningless.

Well...

Consider the Godel sentence for a formal system.  This sentence, on one 
level, merely talks about a particular integer, and so poses no problem 
for the system.  Of course, the real trick in Godel's theorem is viewing 
the kernel of the Godel sentence as an expression in the language of 
the formal system rather than as a "mere" integer.  After all, every 
natural number corresponds to a sentence.

Don't be surprised, however, if the formal system in question retorts:
"Hmph.  You can't slip back and forth between these two levels.  Do 
you want to look at the sentence as being about a mere integer, or 
even about the syntactic properties of some string of words?  If so, 
it places no limits on what I can believe;  I believe _propositions_, 
not strings.  If, however, you want to regard it as about the _content_ 
of some sentence, you're in trouble, for the sentence to whose content 
it is referring is itself, and therefore it's meaningless!"

In fact, I think we have recently seen some formal systems say just that.


-- 
Avrom I. Faderman                  |  "...a sufferer is not one who hands
avrom@csli.stanford.edu            |    you his suffering, that you may 
Stanford University                |    touch it, weigh it, bite it like a
CSLI and Dept. of Philosophy       |    coin..."  -Stanislaw Lem


