So to sum up, while you think linguistic vagueness comes from Godelian
incompleteness, I think Godelian incompleteness can't even be defined
in this context, due to linguistic vagueness.
OK. Personally, I think that you did a good job of defining Godelian
Incompleteness this time but arguably you did it by reference and by
building a new semantic structure as you went along.
On the other hand, you now seem to be arguing that my thinking that
linguistic vagueness comes from Godelian incompleteness is wrong because
Godelian incompleteness can't be defined . . . .
I'm sort of at a loss as to how to proceed from here. If Godelian
Incompleteness can't be defined, then by definition I can't prove anything
but you can't disprove anything.
This is nicely Escheresque and very Hofstadterian but . . . .
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <agi@v2.listbox.com>
Sent: Thursday, October 23, 2008 11:54 AM
Subject: Re: [agi] constructivist issues
Mark,
My type 1 & 2 are probably the source of your confusion, since I
phrased them so that (as you said) they depend on "intention".
Logicians codify the intension using semantics, so it is actually
well defined, even though it sounds messy. But, since that explanation
did not work well, let me try to put it a completely different way
rather than trying to better explain the difference between 1 and 2.
Godel's incompleteness theorem says that any logic with a sufficiently
strong semantics will be syntactically incomplete; there will be
theorems that are true according to the semantics, but based on
allowed proofs, they will be neither true nor false. So Godel's
theorem is about an essential lack of match-up between proof and
truth, or as is often said, syntax and semantics.
To apply the theorem to natural language, we've got to identify the
syntax and semantics: the notions of "proof" and "truth"
that apply. But in attempting to define these, we will run into some
serious problems: proof and truth in natural language is only
partially defined. Furthermore, those "serious problems" are (it seems
to me) precisely what you are referring to.
So to sum up, while you think linguistic vagueness comes from Godelian
incompleteness, I think Godelian incompleteness can't even be defined
in this context, due to linguistic vagueness.
--Abram
On Thu, Oct 23, 2008 at 9:54 AM, Mark Waser <[EMAIL PROTECTED]> wrote:
But, I still do not agree with the way you are using the incompleteness
theorem.
Um. OK. Could you point to a specific example where you disagree? I'm
a
little at a loss here . . . .
It is important to distinguish between two different types of
incompleteness.
1. Normal Incompleteness-- a logical theory fails to completely specify
something.
2. Godelian Incompleteness-- a logical theory fails to completely
specify
something, even though we want it to.
I'm also not getting this. If I read the words, it looks like the
difference between Normal and Godelian incompleteness is based upon our
desires. I think I'm having a complete disconnect with your intended
meaning.
However, it seems like all you need is type 1 completeness for what
you are saying.
So, Godel's theorem is way overkill here in my opinion.
Um. OK. So I used a bazooka on a fly? If it was a really pesky fly and
I
didn't destroy anything else, is that wrong? :-)
It seems as if you're not arguing with my conclusion but saying that my
arguments were way better than they needed to be (like I'm being
over-efficient?) . . . .
= = = = =
Seriously though, I having a complete disconnect here. Maybe I'm just
having a bad morning but . . . huh? :-)
If I read the words, all I'm getting is that you disagree with the way
that
I am using the theory because the theory is overkill for what is
necessary.
----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]>
To: <agi@v2.listbox.com>
Sent: Wednesday, October 22, 2008 9:05 PM
Subject: Re: [agi] constructivist issues
Mark,
I own and have read the book-- but my first introduction to Godel's
Theorem was Douglas Hofstadter's earlier work, Godel Escher Bach.
Since I had already been guided through the details of the proof (and
grappled with the consequences), to be honest chapter 10 you refer to
was a little boring :).
But, I still do not agree with the way you are using the incompleteness
theorem.
It is important to distinguish between two different types of
incompleteness.
1. Normal Incompleteness-- a logical theory fails to completely
specify something.
2. Godelian Incompleteness-- a logical theory fails to completely
specify something, even though we want it to.
Logicians always mean type 2 incompleteness when they use the term. To
formalize the difference between the two, the measuring stick of
"semantics" is used. If a logic's provably-true statements don't match
up to its semantically-true statements, it is incomplete.
However, it seems like all you need is type 1 completeness for what
you are saying. Nobody claims that there is a complete, well-defined
semantics for natural language against which we could measure the
"provably-true" (whatever THAT would mean).
So, Godel's theorem is way overkill here in my opinion.
--Abram
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