Le 26-juin-05, à 08:47, Russell Standish a écrit :

On Fri, Jun 24, 2005 at 03:25:29PM +0200, Bruno Marchal wrote:

Perhaps. It depends of your definition of "OM", and of your
"everything" theory.

Let me tell you the "Lobian's answer": if I have a successor OM then I
have a successor OM which has no successor OM.

OK, I am cheating here, but not so much. As I just said to Stathis I
must find a way to convince people about the urgency of using the modal
logical tools.


This reminds me of something I wanted to ask you Bruno. In your work
you axiomatise knowledge and end up with various logical systems that
describe variously 1st person knowledge, 1st person communicable
knowledge, 3rd person knowledge etc. In some of these, the Deontic
axiom comes up, which if translated into Kripke semantics reads "all
worlds have a successor word" (or "no worlds are terminal").


Or, more simply said: with the logic of of BP, G, the logic of third person self-reference, there are cul-de-sac (terminal world) everywhere. All variants of BP (Theatetus 1, 2, 3) are ways of making abstraction of the cul-de-sac worlds, with the goal of getting probabilities. (And those ways are justified by G* which knows more about the machine, if you remember G*).

To have probability(P) = one, it is enough to have the truth of P in all accessible OMs. But, alas, in a cul-de-sac world/OM we have that Prob(P) = one "trivially" (no counterexemples, I assume classical logic in all worlds).

To have a probability one we must assure the existence of at least one model, (or one accessible OM, or one consistent extension, etc.). This is provide by the deontic transform where DP is defined by BP & ~B~P.

Err ... I hope you remember how to see that ~B~p is equivalent with "p is consistent" for the machine. Modally ~B~P is the dual of BP, it is the diamond which I wrote <>P. "~B~" gives the simple way to talk on "possible world/state/OM..." with the machine.

The machine stays mute if you ask her if there is one (at least) consistent extension (OM). But she becomes chatty when you ask her what would the worlds look like in case some world exists.

Bruno

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


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