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From: George Levy <GLevy.domain.name.hidden>

Date: Sun, 08 Apr 2001 14:41:24 -0700

Marchal wrote:

*> Hi George,
*

*>
*

*> I finish my post yesterday a little too quickly.
*

*>
*

*> I said:
*

*>
*

*> >I hope your binder will not explode.
*

*> >My next post will anticipate toward the end of my proof to glimpse
*

*> >the "quasi-appearance of Hilbert Space" when we explain the UDA TE
*

*> >to the guardian angel of the sound machine.
*

*> >
*

*> >And, just because you promise me a prize for deriving SE from
*

*> >the "psychology of machine" I tell you that I have decided
*

*> >to call the modal formula (the one for the symmetrical frame):
*

*> >
*

*> > p->[]<>p,
*

*> >
*

*> > the little abstract Schroedinger Equation (LASE),
*

*> >
*

*> >as I have called before the (godel-like) formula
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*> >
*

*> > <>p -> -[]<>p
*

*> >
*

*> > the first theorem of machine's psychology. (FTMP)
*

*> >
*

*> >And our goal is to find a natural bridge from FTMP to LASE.
*

*> >
*

*> >I must say that I have believe for a quite long time that this was
*

*> >impossible. But then when you take the definition of knowledge
*

*> >belief, observation/perception in Plato's thaetetus, then the
*

*> >aritmetical translation of the UDA TE will lead us directly toward
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*> >the solution.
*

*> >And so you will be obliged to give me the prize (at least
*

*> >a little abstract price!).
*

*>
*

*> Of course this is not entirely correct. You should "give me
*

*> the prize" only:
*

*>
*

*> 1) when you will see what I mean when I say that
*

*> <>p -> -[]<>p is a fundamental theorem in machine's psychology.
*

*>
*

*> + (above all):
*

*>
*

*> 2) when you understand why I dare to call the formula p->[]<>p the
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*> little abstract schroedinger equation.
*

*>
*

*> + (of course):
*

*>
*

*> 3) when it will be clear how I translate
*

*> the UDA TE and how that translation isolates a derivation of
*

*> LASE from FTMP.
*

As I said in my earlier post, 1) seems inconsistent, 2) seems obvious.

3) I do not recall the acronyms.

Please refresh my memory with regards those acronyms UDA and TE

*>
*

*> I explain informally a little bit.
*

*>
*

*> ABOUT 1):
*

*>
*

*> When the box []p is interpreted in english as provable(p),
*

You are saying that "p is true in all worlds" is identical to "p is

provable." You have lost me....

*> i.e.
*

*> provable by a honest UTM, the <>p, which is -[]-p, can be read as
*

*> p is consistant. This follows from the fact that if -p is not provable
*

*> then you can add p to the set of axioms used by the machine without
*

*> being lead to a contradiction.
*

Great adding an unprovable statement to a set of axiom is OK. But how do

you know it is not provable until you actually attempt to prove it... and

how long will you attempt to do so?

*> (Indeed if the machine derives the
*

*> false from -p, then the machine derives (-p -> FALSE), which is --p,
*

*> which is p). More directly <>TRUE is the same as -[]-TRUE = -[]FALSE,
*

*> = I do not prove the FALSE = I am consistent.
*

*> So <>p -> -[]<>p is just a modal form of a generalisation of Godel's
*

*> second incompleteness theorem, and in English you can read that
*

*> formula in the following way. if p is consistent then I cannot prove
*

*> that p is consistent.
*

*>
*

*> When the machine proves the particular case (<>TRUE -> -[]<>TRUE),
*

*> that is Godel's second theorem, it is as if the machine was telling
*

*> us: if I am consistent I cannot prove it.
*

*>
*

Consistency ("logical thinking") is certainly a necessary condition for

consciousness ("I think" a la Descartes)

*>
*

*> A more psychological reading of that formula is, by identifying
*

*> (audaciously perhaps) consistency with consciousness (or awakeness)
*

*> you get "if I am conscious then I cannot prove it.
*

but consistency is not a sufficient condition for consciousness. Or is it?

What else would be required?

*>
*

*>
*

*> Note that any formula with the form <>p can be read there is an
*

*> accessible observer-moment (world) with p true at it. So when the
*

*> talk about <>p, the machine talk about a consistent extension of
*

*> herself.
*

*>
*

*> To sum up, the 1) above is linked to Godel's theorem, seen as a
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*> psychological limitation of machine. Later I will be hopefully
*

*> a little more rigorous about the link consciousness/consistency.
*

*>
*

*> ABOUT 2):
*

*>
*

*> Why do I consider the formula p->[]<>p as an abstract form
*

*> of SE ?
*

*> Surely that deserves some words of explanation. Unfortunately
*

*> to explain this, we must leave the cocooning logic of plato
*

*> heaven (classical logic) for the jungle of what is called
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*> the WEAK LOGIC.
*

*> The theorems of a weak logic makes a subset of the theorems
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*> of classical logic.
*

*>
*

*> We will meet essentially two principal weak logics:
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*> intuitionistic logic and quantum logic.
*

*>
*

*> For exemple a typical classical tautology which is not a theorem
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*> for intuitionnistic logic is the principle of excluded middle
*

*>
*

*> p v -p
*

*>
*

*> And a classical tautology which is not a theorem of quantum logic
*

*> is
*

*> p & (q v r) <-> (p & q) v (p & r)
*

*>
*

*> That is, in quantum logic there is a failure of the distributivity
*

*> axioms.
*

*> It has been proved that both intuitionist and quantum logic
*

*> doesn't have truth table semantics. But modal logics can help
*

*> us to handle those weak logics without leaving plato heaven.
*

*>
*

*> In particular the modal logic, known as B, with the axioms
*

*> []p ->p, p->[]<>p and with the rule of modus ponens and necessitation
*

*> has a non trivial relationship with quantum logic.
*

*> In fact we have a representation theorem (by Goldblatt) which
*

*> says that when B proves []<>A, QL proves A. (I simplify a
*

*> little bit). Sometimes []<>A is called the quantization of A.
*

*> Isolating the formula p->[]<>p is also a good step toward
*

*> defining a measure on the relative set of consistant extension.
*

*>
*

*> ABOUT 3). (later).
*

*>
*

*> I stop here. I hope you are not overwhelmed by the information.
*

Yes I am.... but as the saying goes "l'appetit vient en mangeant." We

shall see.

Forgive me for not being too prompt....Responding to your posts is not

simply a matter of making a few cute remarks....We are certainly going in

the right direction....I think that the basic ideas are probably very

simple... the hard part is to communicate them.

*>
*

*> Here I have just try to anticipate a little. I must say that I
*

*> not to busy now, but unfortunately I will be quickly more and more
*

*> busy. So I give you something to eat for a long period.
*

*> (except that I will try to say a little more about 3) still
*

*> this week).
*

*> Nevertheless, don't hesitate to tell me "enough!" or one of your
*

*> "hmmmmmmmmmmmmmmmmmm". I would understand. Of course it is not
*

*> so easy to understand my work which is not only transdisciplinar
*

*> but also goes accross very different kind of logics.
*

*>
*

*> Bruno
*

George

Received on Sun Apr 08 2001 - 14:44:42 PDT

Date: Sun, 08 Apr 2001 14:41:24 -0700

Marchal wrote:

As I said in my earlier post, 1) seems inconsistent, 2) seems obvious.

3) I do not recall the acronyms.

Please refresh my memory with regards those acronyms UDA and TE

You are saying that "p is true in all worlds" is identical to "p is

provable." You have lost me....

Great adding an unprovable statement to a set of axiom is OK. But how do

you know it is not provable until you actually attempt to prove it... and

how long will you attempt to do so?

Consistency ("logical thinking") is certainly a necessary condition for

consciousness ("I think" a la Descartes)

but consistency is not a sufficient condition for consciousness. Or is it?

What else would be required?

Yes I am.... but as the saying goes "l'appetit vient en mangeant." We

shall see.

Forgive me for not being too prompt....Responding to your posts is not

simply a matter of making a few cute remarks....We are certainly going in

the right direction....I think that the basic ideas are probably very

simple... the hard part is to communicate them.

George

Received on Sun Apr 08 2001 - 14:44:42 PDT

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