From: Marchal <>
Date: Tue Mar 20 02:33:20 2001

George Levy wrote:

>It shows that COMP entails the general shape of SE but not SE proper.

We will appreciate this later.
UDA entails ``necessary (COMP entails SE proper)",
independently of the fact that perhaps only 10^100 working
mathematicians would be needed during 10^100 years for finishing the
derivation. But yet, philosophically it is a point that COMP entails
that. Now, it happens I isolate with my UTM interviews a
non trivial part of the derivation, perhaps. You will judge.

>> [BM] I also use the COMP hypothesis which help by giving both TEs, and
>> precise mathematical theories. It is frequent that focusing on point
>> and focusing on links are related by dualities. With the Z logics the
>> links are really topological. I think I have shown a way to derive
>> the thickness from the turing-tropic views.
>I don't understand, but if you actually found a way to derive the
>"thickness" (
>which I guess is Planck's constant) from purely philosophical arguments, you
>deserve a Nobel Prize. However, until I see the proof, I strongly doubt
>that you
>have achieved such a feat.

OK, same answer. I will show you the proof.

>Right, I think we are now in agreement! The key words are CONSISTENT
>This means that a consciousness can be fuzzy... and the world it lives
>in... or
>rather each world (in the whole set of worlds it lives in) must be a
>extension of this consciousness. The degree of the consciousness fuzziness
>corresponds to the size of the set of worlds. The word "consistent" implies
>logical consistency where logic is in relation to the consciousness of the


>> [BM] Turing-tropic instead. That the simple shift I do. There is a nice
>> mathematical way to interview the self introspecting honest machine.
>> (BTW and by Solovay there are *two* nice utterly non trivial ways).
>I don't understand Turing-tropic

Mhh... the fact is that we can interview Sound Universal Turing
Machines about themselves and about their consistent extensions.
It is not easy. Godel's theorem and Lob's theorem is just a beginning
of that interview. Then Solovay find axiomatic systems ...
I will explain.

>No they don't share life stories. They share *memories* of life stories.
>true or false are PRESENT properties of their mind. This is why I rather
>assume a
>first order Markov process, which does not depends on past information.

At the substitution level we can certainly be described by such a first
order Markov process.

But with comp we cannot know for sure the substitution level (this
is Benaceraff's insight: if we are consistent machine we cannot know
consistently which one we are; this is of course linked to the godelian
"fuzziness" of the set of our consistent extensions.

But perhaps there is something more I should ask you before. You said
in response to some post of me, in some preceeding dialog:
    <<<I smell a whiff of third person thinking.>>>

Well, I know you are not stuck in third person (like Schmidhuber and
Mallah) but I hope you are not exclusively first-person.

It is useful to identify 3-person honest communication with
belief (proof). let us write []A for: the sound machine
can prove (believe) A.

It is natural to have

                       [](A->B) -> ([]A ->[]B)

meaning that if it is provable that (A->B) then, if it is provable
that A then it is provable that B.

It is useful to identify the first person with the knower, and, if I
write [.]A for I can know A, it is natural to have also:

                       [.](A->B) -> ([.]A ->[.]B)

Both []A -> [][]A and [.]A -> [.][.]A are quite natural once we
treat sufficiently introspective machine or entities.
For them: A is believable entails that it is believable
that A is believable, and the same for knowable.

It is natural to have that the knowability entails the believability:
That is: [.]A -> []A. Do you agree ? It just mean that if A is knowable
then A is believable.

OK ?

With the intuitive meaning of belief and knowledge the reverse
is certainly wrong: the fact that A is believable does not entails
that A is knowable. We don't have []A -> [.]A. For exemple it is
believable that the earth is flat, but that is hardly knowable.

In fact it seems that for knowledge we have [.]A -> A,
but for belief we don't have []A -> A.

Would you basicaly agree until here ...

It looks like I propose you some analytical philosophy.
Later I will propose you *arithmetical* philosophy. The meaning
of [] and [.] and others toward observability will be defined
in arithmetic and only then will the UDA be translated in
that arithmetical philosophy and then the necessarily hilbertian
structure on the consistent extensions will begin to appear.

Do you know some elementary classical logics?
The magic which will put an unexpected and rather involved
fuzzy structure on the set of consistent extensions comes
from godel's theorem (with [] modelize by formal provability).
It looks like an oversimplification, but even with that
oversimplification we will get non trivial theories of belief,
knowledge, observation, etc.

Do you know Aristote Modal Square: (read []p "necessarily p")
"-" is not.

                            []p []-p

                           -[]p -[]-p

or its dual (read <>p possibly p):

                           -<>-p -<>p

                            <>-p <>p

They match. I mean []p is equivalent to -<>-p. ``It is necessary
that p" is equivalent with ``it is not possible that -p", and
``it is possible that p" is equivalent with "it is not necessary
that -p". And then []-p is equivalent with -<>p, and <>-p is
equivalent with -[]p.

Later []p will mean p is formally provable, then <>p is read as
To say p is consistent is indeed equivalent with saying that -p
is not provable.
Consistency, that is the non provability of FALSE, is then
equivalent with the consistency of TRUE, and Godel's theorem
can be written:

                       <>TRUE -> -[]<>TRUE.

TRUE and FALSE are just two propositional constant. You can
identify TRUE with the proposition (1 = 1) and FALSE with the
proposition (1 = 0).
We are just at the beginning. In a next post I will explain
Leibnitz semantics. It is fun. And it will help us for
introducing Kripke semantics, which is just a "relativisation"
of Leibnitz semantics. That will even, perhaps, help us
to understand the difference between the absolutist and the
relativist in our discussions.

Received on Tue Mar 20 2001 - 02:33:20 PST

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