Re: White Rabbits and QM/Flying rabbits and dragons

From: Marchal <marchal.domain.name.hidden>
Date: Fri Nov 26 08:48:25 1999

Russell Standish wrote:

>> I guess you are talking about usual senility. That argument doesn't
>> convince me. More about that later, when I will explain more clearly
>> the 1-white rabbit through the UDA.
>>
>
>No I'm not. I was actually referring to a comment I made on QTI
>earlier, when I stated that the universe will become increasingly
>unlikely as we age. For example, as our "measure" - in the Jacques
>Mallah sense decreases as we start to live longer than our natural
>lifespan, increasingly more improbable thing must happen to keep us
>alive. Examples mentioned were things like wormholes, uploads into
>different hardware etc.

I remember. I told you it was whishfull thinking.
I don't fear White Rabbit, you know. But what about the Snark ?
And what if the Snark *is* a Boojum? You see?


>I'm having problems interpreting comp_n as consistency (it is not how
>it is defined).

You are right. I'm used to modelize comp_n not by the consistency
but by the consistency of consistency. But this doesn't change
the math. In term of Kripke model consistency means there is
at least one word I can access. In the drawing (!) T is the
propositional constant TRUE.

 ------------------ ------------------
 I I I I
 
 I /\ ----- I \ I ----- I
 I / \ I I---------------------I I I
 I \ / I I / I I I
 I \/ I I I I I
 I I I I
 ------------------ ------------------

>The I have trouble interpreting consistency as
>possibility (the \Diamond symbol).

Take a formal theory like PA. To say that a proposition P
is consistent with PA means that PA does not prove the
negation of P. So you can add P to extend consistently PA.
When the box is interpreted as formal
provability then consistency of P is NOT BOX NOT P, which is
DIAMOND P.
Simple consistency is consistency of the propositional constant TRUE.
It means that there is an extension of the theory where TRUE is true,
which is trivial if the theory cannot prove FALSE, i.e if the theory
is consistent.

>> 2) The second roots is the duplication part of the UDA. Read the box
>> as 'I can communicate scientifically that'
>
>How do I do that?

Eh ... When you see "BOX P", just read ``I can communicate
scientifically that P". You can modelize scientific communication
by classical formal provability in an axiomatizable theory. It is
a typical third person theory. Thanks to Solovay such communication
is entirely formalised by BOX (BOX P -> P) -> BOX P (Lob's formula)
in a normal system of modal logic.
Of course there are a lot of systems of modal logics. If you read the
box as "always", then you read the diamond as "sometimes" and of
course you will need other axioms to formalise that.
You can even formalize the notion of ``informal proof". Typically
you will need an extension of the system S4 (see Chellas).
In deontic logic the box is read as "obligatory that" and the
diamond is read as "permitted that", a typical axiom is
``BOX A -> DIAMOND A", because it is unfair to make something
obligatory, yet not permitted !

By a kind of generosity of my part ;-), I have made publicity for the
book on Modal Logic by Chellas because it is one of the best book
on modal logic, and I think modal logic can help all everythingers.
But, as I said sometimes ago, for my thesis, you should read the
book by Boolos 1993: the logic of provability. If you are lucky
you can perhaps still find the preceeding book by Boolos, which
is "The Unprovability of Consistency" which is more readable and
has a lot of charm. Both books are published by Cambridge University
Press. There is also the nice complementary book by Smorinski :
"Modal Logic and Self-Reference".
And remember the delightfull little introduction to the modal logic
of formal provability by Smullyan: "Forever Undecided". It exists
in paperback.

>> In that case if I duplicate at some level n a person which does not
>> believe in comp_n, then the doppelganger will understand that he will
>> never be able to communicate (scientifically) to the original person
>> that `he' has survived the duplication. That is comp_n is not
>> scientifically communicable.
>
>In what sense is it not scientifically communicable? I thought it was
>just done so in your thesis - regardless of whether one believes in it.

You should carefully distinguish between COMP which is that there is a
level n such that I survive a functionnal substitution made at the level
n. And COMP_n, which is that I survive at the level n. That distinction
is important because:

``BOX(Exists n COMP_n)" does not imply ``Exists n BOX (COMP_n)".

The fact that "I prove there is a level such that I survive a substitution
at that level" does not entail that "there exists a level such that I
can prove I will survive the substitution at that level".
This is because I don't ask
the constructivity of the notion of provability (it is *classical* proof
theory, not intuitionistic or constructive proof theory). So we cannot
in general permutate quantifiers and BOX.

>In hope of finding an understanding of all this stuff...

It is just a question of time and motivation. I think.
Perhaps I would like to suggest
not embarking yourself to quickly in the formal logical approach.
The meaning of what I try to do really comes from the argument that
Physics must be a branch of Machine Psychology, i.e. the ``reversal"
consequence of the UDA. I admit this is a rather counter-intuitive
result.
Then the modal tools appears to provide a big help for putting order
on what is left after the reversal.
Of course my chapter 5 is at the intersection of the foundation of QM, the
Godelian Self-reference logics, philosophy of mind and theoretical
computer science. It cannot be easy. But the really fundamental
step is the UDA (chapter 3) which needs only passive familiarity
with numbers and computers, andit provides the motivation for
the formal approach.
...Well UDA needs also a minimal amount of folk-psychology for the needed
partial understanding of the word "survive". Then you can look at the
chapter 5 as an eventual elimination of that folk-psychology in UDA. In
the chapter 5 folk-psychology is substitute by the logics (plural) of
self-reference.

Bruno




 
Received on Fri Nov 26 1999 - 08:48:25 PST