From newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!rpi!psinntp!psinntp!scylla!daryl Tue Nov 24 10:52:06 EST 1992
Article 7645 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
Path: newshub.ccs.yorku.ca!torn!cs.utexas.edu!zaphod.mps.ohio-state.edu!rpi!psinntp!psinntp!scylla!daryl
>From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <1992Nov14.134537.2170@oracorp.com>
Organization: ORA Corporation
Date: Sat, 14 Nov 1992 13:45:37 GMT
Lines: 96

In article <1992Nov12.212403.32326@Cookie.secapl.com>,
frank@Cookie.secapl.com (Frank Adams) writes:

>>Statement G:
>>
>>     `Diagonalizing `Diagonalizing this sentence produces a string of words
>>      that will never be believed by David Chalmers.' produces a
>>      string of words that will never be believed by David Chalmers.'
>
>By the way, this statement as is can easily be falsified by David Chalmers.
>All he has to do is believe *once* -- thereafter he, and everybody else, can
>consistently recognize it as false.

Exactly. G is true if and only if David Chalmers does not believe G.
If David Chalmers *does* believe G, then G is false (and David
Chalmers happens to believe a manifestly false statement).

The fact that G *can* meaningfully be false shows that G is not a
paradoxical statement. And it can meaningfully be true, as well.
It all depends on whether David Chalmers' beliefs are consistent.

>>>And as a statement, G is self-referential in an unacceptable way.  Consider
>>>sentence H:
>>>
>>>`Diagonalizing `Diagonalizing this sentence produces a string of words which
>>>is false when interpreted as a statement.' produces a string of words which
>>>is false when interpreted as a statement.'
>>>
>>>If we can interpret G as a statement, we can interpret H likewise; but this
>>>produces a flat-out contradiction.  Thus it is *not* legitimate to interpret
>>>G as a statement, however clear its meaning may seem.
>>
>>I disagree. The problem with statement H is *not* its self-reference.
>>G and H both have the same form, roughly:
>>
>>     G <-> not(G in B)
>>     H <-> not(H in T)
>>
>>where B is the set of believed sentences, and T is the set of true
>>sentences. The problem is *not* that G or H is self-referential, it is
>>the fact that there is no consistent notion of the set of all true
>>sentences. As a matter of fact, your sentence H shows that there can
>>be no such set.
>
>Wait a minute!  Where did these sets come from?  I didn't commit myself to
>the existence of any particular sets.

I don't care whether you committed yourself to anything; I was simply
trying to give an analysis of a statement like "This sentence is
false" (roughly sentence H above). You claimed that the problem with
such sentences is self-reference. I think that is nonsense;
self-referential sentences do not cause any particular problem when
the predicates the sentences talk about are well-defined, for example:

"This sentence is six words long."
"This sentence is written in English."

However, the notion of truth and falsity, if applied in an
unrestricted way, *does* result in a contradiction. This is easily
seen in the classic "This sentence is false" or paradox "The smallest
number that cannot be defined in less than twenty words." (The latter
is a paradox because it seems to define a number in less than twenty
words.)

>Even so, you have done nothing to argue that there *is* a set of
>believed sentences -- you have just failed to derive a contradiction
>from the assumption that there is.

Fine.

>Note the contrast with provability for a formal system.  G"odel's sentence
>can be shown to be equivalent to provability in the formal system.  By
>contrast, you are blandly asserting the right to do so with the word
>"believe".

I didn't blandly assert it. I constructed the sentence G so that it
has this property. If you would read Godel's proof, you will find the
construction almost identical; the only differences are (1) instead of
the predicate "is believable", he uses the predicate "is provable" (2)
instead of coding statements by strings, he codes them by numbers.
Godel's proof is much more difficult because neither the provability
predicate nor the diagonalization operation exists as primitives in
the language of Peano arithmetic, so they have to be pains-takingly
built up from the primitives +,*,=, and the logical operators.

>I still believe that sentences like G are invalid.

Invalid in what sense? Is the notion of diagonalizing a string
invalid? Is the notion of David Chalmers believing something invalid?
G is a simple combination of these two notions.

Daryl McCullough
ORA Corp.
Ithaca, NY




