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From: Jacques Bailhache <jacquesbailhache.domain.name.hidden>

Date: Fri, 10 Sep 2004 09:31:40 +0000

Hi Bruno,

*>From: Bruno Marchal <marchal.domain.name.hidden>
*

*>To: everything-list.domain.name.hidden
*

*>Subject: Re: Lob + New Views On Mind-Body Connection
*

*>Date: Thu, 09 Sep 2004 15:46:10 +0200
*

*>
*

*>Hi Jacques,
*

*>
*

*>Nice to see you back. Actually I just discovered your message
*

*>in the archive, I did not got them by the mail (?). Sorry for the delay.
*

*>I quote you from the archive:
*

*>
*

*>
*

*>
*

*> >The axiom B(Bp->p)->Bp seems very strange to me.
*

*>
*

*>I think it *is* strange. It is at the heart of "counter-intuition" in the
*

*>sense
*

*>that you can derive from it (together with K = B(p->q)->(Bp->Bq) and the
*

*>two inference rule MP and NEC) all the consequences of incompleteness.
*

What is the NEC rule ?

*>Do you see how to derive Godel second theorem of incompleteness theorem?
*

I think I see how to derive Gödel theorem : if we take p=F (false) we get

B(BF->F)->BF or B(~BF)->BF. BF means that the machine is inconsistent, ~BF

that it is consistent. If C means that the machine is consistent, then

B(~BF)->BF becomes BC->~C, which means that if the consistency is provable,

then the machine is not consistent.

*>Boolos gives 5 reasons to find Lob's formula, that is B(Bp->p)->Bp,
*

*>"utterly astonishing", and he does not mention the placebo effect.
*

*>(have you read my last paper?
*

*>http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHAL.htm )
*

I had a look at it.

*>
*

*>
*

*> >Is it applicable only to machines or also to humans ?
*

*>
*

*>
*

*>It is applicable to any consistent believer in
*

*>(classical) arithmetic, when belief are
*

*>checkable. If you prefer the B is for
*

*>provable, or Beweisbar (Godel).
*

*>For any machine-like entity (with or without
*

*>oracle) it gives even their complete
*

*>(propositional)
*

*>logic of provability and consistency.
*

*>It *is* the main axiom of G.
*

*>(Note that there is a corresponding
*

*>version for intuitionist arithmetic.)
*

*>
*

It is more clear to me if B means "provable" rather than "believe".

But I wonder if the notion of provability is equivalent to the notion of

belief. Intuitively I have the impression that if I don't believe that Santa

Claus exists, then I believe that I don't believe that Santa Claus exists.

One can believe sonething without having a proof of it. If it is a checkable

belief, why not say that the machine is sure that p, rather than believes p

?

*>
*

*>
*

*> >I don't believe that Santa Claus exists (~Bp).
*

*> >If I consider the proposition "Bp->p" which
*

*> >means "If I believe that Santa Claus exists,
*

*> >then Santa Claus exists", this proposition
*

*> >seems true to me, because of le
*

*> >propositional logic rule "ex falso quodlibet
*

*> >sequitur" or false->p : the false proposition
*

*> >Bp implies any proposition, for example p.
*

*> >So I can say thay I believe in the proposition
*

*> >Bp->p : B(Bp->p). According to the lobian
*

*> >formula B(Bp->p)->Bp, this implies Bp (I believe
*

*> >that Santa Claus exist) !
*

*>
*

*>It is all correct .... except that you cannot
*

*>prove (believe) your own consistency; so that
*

*>you cannot prove that you don't believe that Santa
*

*>Klaus does not exist. All formula beginning by ~B
*

*>are not believable (provable) by the consistent machine.
*

*>
*

*>
*

*>
*

*> >More formally : The axiom ~Bp->B(~Bp)
*

*> >seems correct to me, isn't it ? <snip>
*

*>
*

*>
*

*>It seems, but not for a notion of checkable or
*

*>verifiable belief. Any machine capable of proving
*

*>there is a proposition she cannot prove, would
*

*>be able to prove its consistency, and that's impossible
*

*>by Godel II. (this follows from your "ex falso quodlibet":
*

*>a machine proving the f, will prove all propositions, so if
*

*>there is a proposition (like Santa Claus exists) that
*

*>you pretend you will never believe (prove) then you
*

*>are asserting that you prove (believe) you are consistent!.
*

*>
*

*>All right? I use "believe" instead of "prove" because
*

*>we are following a little bit Smullyan's "Forever Undecided".
*

*>But this suits well with the "machine psychology".
*

*>Do you have that FU book?
*

I don't have this book.

*>
*

*>Bruno
*

*>
*

*>
*

*>http://iridia.ulb.ac.be/~marchal/
*

*>
*

Jacques.

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Received on Fri Sep 10 2004 - 05:38:19 PDT

Date: Fri, 10 Sep 2004 09:31:40 +0000

Hi Bruno,

What is the NEC rule ?

I think I see how to derive Gödel theorem : if we take p=F (false) we get

B(BF->F)->BF or B(~BF)->BF. BF means that the machine is inconsistent, ~BF

that it is consistent. If C means that the machine is consistent, then

B(~BF)->BF becomes BC->~C, which means that if the consistency is provable,

then the machine is not consistent.

I had a look at it.

It is more clear to me if B means "provable" rather than "believe".

But I wonder if the notion of provability is equivalent to the notion of

belief. Intuitively I have the impression that if I don't believe that Santa

Claus exists, then I believe that I don't believe that Santa Claus exists.

One can believe sonething without having a proof of it. If it is a checkable

belief, why not say that the machine is sure that p, rather than believes p

?

I don't have this book.

Jacques.

_________________________________________________________________

Bloquez les fenêtres pop-up, c'est gratuit ! http://toolbar.msn.fr

Received on Fri Sep 10 2004 - 05:38:19 PDT

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