Dear Stephen,
Kim Jones' post prompts me to ask whether or not a
"self-referentially-correct Loebian machine" involves an infinite
regress or a non-well founded structure. Given that it is typical to
include the idea of a non-prescripted interview, where the questions
can have follow ups based on answers given and thus not prespecified,
how does a Loebian machine prevent a pathological regress? Is this
where one is really coming up with a fancy secular notion of
"omniscience" (infinite computational/simulation power)?
Any idea?
It depends what do you mean by "Omniscient", a word which has different
meaning in Applied Logic in AI, Mathematical Logic, Epistemology, and
theology.
If by omniscience (of an entity) you are meaning the knowability of all
what is knowable, then there is a form of omniscience, of the kind
avoid in Artificial Intelligence. This comes from our interest in
provABILITY, and knowABILITY, etc. The logiocal consequence of the
entities belief/knowledge are believable (knowable).
But this is different from knowing all the truth (by incompleteness).
Here G* can, at the propositional level, be considered as omniscient,
but then remember that can G is not a reasoner reasoning about itself.
Well-foundation is asked for the provability, and the Lob formula
B(Bp->p)->Bp can somehow be interpreted as a form of induction axiom,
or as a well fondation axiom for the structure of the proof given by
the machine/entity. But this has nothing to do with the belief of the
machine proper. The basic belief can be the axiom of Peano Arithmetic,
or of some Set Theory with an axiom of foundation, or without or with
an axiom of antifoundation, etc.
The self-reference of the lobian machines and the lobian angels does
not imply infinite regress. Actually it is the original Godel's point
that such self-references can be done in a finite way, with finite
means, without leading to paradoxes or contradictions. The trick is
based on the diagonalization procedures of applying a copying machine
to itself. I try to illustrate this sometimes by a little song/puzzle:
if DA gives AA, and DB gives BB, and DC gives CC, what does give DD ?
Best regards,
Bruno
Onward!
Stephen
----- Original Message ----- From: "Kim Jones"
<[EMAIL PROTECTED]>
To: "uv" <[EMAIL PROTECTED]>
Cc: "Bruno Marchal" <[EMAIL PROTECTED]>; "Everything-List List"
<everything-list@eskimo.com>
Sent: Friday, February 17, 2006 7:05 PM
Subject: Re: belief, faith, truth
Which is very interesting, isn't it? People do seem want the kind of
modelled structure for their existence that theology projects. Even
though G means we can never know the truth of it, theology tells us
it is nonetheless there.
Has anyone on this list read Neale Donald Walsch's "Conversations
with God?" series of books? Bruno may well be interested to read at
least Volume 1 if he hasn't yet encountered it. The whole book IS the
interview with the self-referentially-correct Loebian machine! I
realised this yesterday after re-reading sections of it and comparing
them to Bruno's thinking.
http://iridia.ulb.ac.be/~marchal/
