Le 20-juil.-06, à 13:46, Russell Standish a écrit :
> Bruno, I know in your theory that introspection is a vital component
> (the Goedel-like constructions), but I didn't see how this turns back
> onto the self-awareness issue. Did you develop this side of the
> argument?
Yes sure. The Goedel-like construction can handle only a 3-person
discursive self-reference.
A little like if you where reasoning on some 3-description of your
brain or body with your doctor, although it could be also an high level
3-description (like I have a head).
The Goedel construction leads to the modal logic G, where the atomic
propositions "p" are interpreted by arithmetical sentences, and the
modal box "B" by the Goedel provability predicate. So, for example,
with f the logical constant FALSE, and "~" = negation, Goedel second
incompleteness can be written
~Bf -> ~B ~Bf
you can read like the machine says that if the false is provable by er,
then that very fact (that the false is not provable) is itself not
provable.
But "B" is only one among many notion of person, and G corresponds to
an intellectual, discursive, 3-describable, scientific if you want,
sort of self-referential discourse. G* also, but G* got the whole truth
(at this propositional level thanks to Solovay theorem).
Consciousness and first person self-awareness will correspond to
"theaetetical weakening" of that Goedlian provability notion, besides
the arithmetical version of comp.
Mainly:
Bp & p
Bp & ~B~p
Bp & ~B~p & p
or if you abbreviate ~B~p by the diamond: Dp,:
Bp & p
Bp & Dp
Bp & Dp & p
Although G* can prove the arithmetical equivalence of those
"provability predicates", G cannot prove them equivalent, and they will
give rise to different modal logics corresponding to different internal
perception of number-truth. Bp & p gives a temporal knower in an
branching multiverse, Bp & Dp gives rise to a quantum sort of sharable
credibility or a bottom symmetrical multiverse, etc.
Actually, thanks to the nuances you inherit from the corona G* \ G,
those "hypostases" are divided into sharable and no sharable part
making quanta particular case of qualia.
I will come back on the correspondence later. The key point is that the
nuance between
p, Bp, Bp & p, Bp & Dp, Bp & Dp & p, are imposed by the incompleteness
phenomenon, and self-awareness corresponds to the one having " & p" in
their definition. It is the umbilical chord between "truth" and
intellect of the "reasonable" first person.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Sat Jul 22 2006 - 10:51:13 PDT