Le 03-déc.-05, à 11:06, Russell Standish a écrit :
On Mon, Nov 21, 2005 at 03:39:58PM +0100, Bruno Marchal wrote:
Observation is implicitly defined here by measurement capable of
selecting alternatives on which we are able to bet (or to gamble ?).
The french word is "parier".
Well at least this isn't a problem of translation. But I still have
difficulty in understanding why Pp=Bp & -B-p should be translated into
English as "to bet on p" (or for that matter pourquoi on devrait
le traduire par "a parier a p")
For me Bp & -B-p is simply a statement of consistency - perhaps what
we mean by mathematical truth.
~Bf, which is equivalent to D~f, or Dt can be considered as a
consistency statement in case "B" represents some "provability" notion.
Indeed ~Bf = NOT PROVABLE FALSE, and by definition a machine is
consistent if the machine does not prove the false.
And when we will "interview" the Lobian machine, "B" will indeed denote
some provability-by-the lobian-machine notion.
But here we were in a somehow more abstract (thus more easy!)
presentation, which at this stage let completely open how "B" will be
interpreted. In that case you can also consider the formula ~Bf, or Dt,
as a consistency statement, just a more abstract one.
Now in term of a Kripke frame/multiverse: Dt means "I am alive", or "I
am in a transitory state", or "I have access to at least one accessible
world", etc.
More generally ~Bp (or D~p) is a stronger "consistency statement"
meaning that I cannot prove p, meaning that there is an accessible
world where ~p is true.
Now, Bp & ~B~p, that is Bp & Dp, is a much stronger statement saying
that not only p is consistent or possible, but that p is also
"provable/necessary/", which in multiverse term, means that p is true
in all accessible worlds.
So Bp means (in some world alpha) "p is true in all accessible (from
alpha) worlds". Note that if B represents some provability predicate
written in first order logic, then by the most fundamental COMPLETENESS
theorem of Godel (1930, one year before his incompleteness result) it
can be shown that Bp is true if and only if p is true in all the model
of the theory/machine. So Bp is *the* natural candidate for asserting
that "p has probability one", given that Bp means "p is true in all
accessible world".
But now, by the second incompleteness theorem, the machine cannot prove
that Bp -> Dp, because that would imply Bt -> Dt, and, giving that Bt
is provable, this would entail Dt is provable, but for sound lobian
machine Dt -> ~BDt, that is "if I am consistent then I cannot prove my
consistency".
In term of (arbitrary) multiverse, it is even simpler: we just could be
in a cul-de-sac world, where Bf is always true, and Dt is always false,
and clearly this shows that Bp cannot, in general, be taken for
"probability of p is equal to 1": we need to add explicitly the
assumption that there is at least one accessible world!
So "probability of p (in world alpha) is equal to one" is well captured
by Bp&Dp (in world alpha). This means (Kripke-semantically) "p is true
in all accessible world & there is at least one possible world where
true is false".
Of course G* knows that Bp is actually equivalent with Bp & Dp, but the
machine has no way to know that, so, from the machine's point of view,
the logic of the new box B'p defined by Bp & Dp, will be a different
logic. Exercise: show that B'p -> D'p.
And then, if p is verifiable or just attainable by the universal
dovetailer, then it can be shown that p obeys to p->Bp, and this leads
B'p to a quantum logic. The" probability 1" pertaining to the
"provable-and-consistent" verifiable (DU-accessible) proposition gives
a non boolean quantum logic.
Tell me if this is clear enough. Euh I hope you agree that "To bet on
p" can be used for the probability one, of course. If that is the
problem, remember I limit myself to the study of the "probability one"
and its modal dual "probability different from zero".
I must go now and I have not really the time to reread myself, hope I
manage the "s" correctly. Apology if not. Please ask any question if I
have been unclear.
Bruno
http://iridia.ulb.ac.be/~marchal/
