From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Organization: ORA Corporation
Date: Tue, 17 Nov 1992 15:27:53 GMT

frank@Cookie.secapl.com (Frank Adams) writes:

>>>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.
>
>You missed my point -- I was complaining about your wording.  Specifically,
>the phrase "will never be believed".  Mr. Chalmers need only only believe
>G momentarily in order thereafter to have a consisent view of its truth.

I know. I specifically worded it so that it has that property. If
David Chalmers even once believes G, then G is false.

>But this is all by the way, since I think G is NOT meaningful.

You just said that G could be false. How can it be false if it is
not meaningful?

>I am quite familiar with G"odel's proof; also the relevant paper by
>Smullyan.  The point is that in G"odel's proof, he can construct the whole
>machinery formally, and then put the interpretation on it.  Although you
>claim to have done this, you have failed (IMO).

The reason that Godel has to work so hard to construct the machinery
is because the notions of "diagonalization" and "provable" are not
primitives in the language of arithmetic; they have to be tediously
built up from the primitives of +,*,0, successor, =, >, and the logical
operators.

On the other hand, humans use a much richer language, belief is more
or less a primitive, and diagonalization is easily defined.

>>>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.

>The notion of David Chalmers believing something is only valid if the
>*something* is meaningful.  Does he "believe" or "disbelieve" the sun?

It is obviously true that David Chalmers does not believe `the sun'.
He can only believe things which he can understand to be statements.
However, not believing something does not imply that one "disbelieves
it"; to disbelieve something usually means to believe that the
negation is true.

It is a perfectly meaningful statement (and true) to say that `David
Chalmers does not believe `the sun''.

Perhaps I need to clarify things: When I say that David Chalmers
believes a string S, I mean that (1) David Chalmers finds that S
expresses something meaningful, and (2) David Chalmers finds that the
statement expressed by S is true. Otherwise, David Chalmers does *not*
believe S. It is probably the case that David Chalmers does not
believe `flippity gupples are goo', since he probably would not find
that meaningful. It is probably also the case that David Chalmers does
not believe `flippity gupples are not goo'. 

I repeat: it is perfectly meaningful to say David Chalmers believes G,
even when G is nonsensical (although in that case, it will be probably
be false that David Chalmers believes G).

>When you try to understand what G means, by interpreting the "diagonalizing"
>operator, you get an infinite regress (in fact, a simple self-reference, but
>it is not difficult to construct examples using diagonalization where the
>references are not self-referential, but do produce an infinite regress).

There is no infinite regress! To understand what G means, you only
need to know (1) what diagonalization means, and (2) what it means for
David Chalmers to believe something. We know what it means for David
Chalmers to believe G even when G is nonsensical: David believes G if
and only if he finds G meaningful and he thinks that G truthfully
describes the state of the world. If he finds G nonsensical, then he
does *not* believe G. If he does not believe G, then G is true! Proof:

1.     G_0 is the sentence `Diagonalizing this sentence produces a sentence
       that will never be believed by David Chalmers'.

2.     G is the sentence `Diagonalizing `Diagonalizing this sentence
       produces a sentence that will never be believed by David Chalmers' 
       produces a sentence that will never be believed by David Chalmers'.

3.     David Chalmers finds G meaningless (assumption)

4.     Therefore, David Chalmers will never believe G (he can't believe
       things that he finds meaningless).

5.     Since diagonalizing G_0 produces G, and David Chalmers never believes
       G, then we conclude: Diagonalizing G_0 produces a sentence that will
       never be believed by David Chalmers.

6.     Substituting the definition of G_0 into 5, we conclude: Diagonalizing
       `Diagonalizing this sentence produces a sentence that will never be 
       believed by David Chalmers' produces a sentence that will never be
       believed by David Chalmers.

7.     We note that the conclusion of 6 is G itself, so we have a proof of G.

Therefore, the assumption that G is meaningless, and David Chalmers never
believes meaningless things leads to the conclusion that G is true.

>Note that "This sentence has six words." has a different status, because its
>correctness does not involve a recursive reference to its meaning.

That is a very important point, but it also applies to the sentence G.
To determine whether G is true, we need not recursively ask what G
means, we only need to ask whether David Chalmers believes G. That
ends the recursion, since in any case we can't tell whether David
Chalmers believes G solely on the basis of the meaning of G; we need
to know the mechanism by which David Chalmers comes to believe things.
But if we knew the mechanism, then we still would simply ask whether that
mechanism will ever come to believe G; we don't need to know the meaning
of G to determine this, all we need is the syntactic form of G, which is
the same as in the case of "This sentence has six words".

The only way that G becomes paradoxical is if we assume that David
Chalmers is so smart that he believes something if and only if it is
true. In that case, G reduces to the Liar Paradox, which I admit does
cause problems.

Daryl McCullough
ORA Corp.
Ithaca, NY

























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