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