Le 05-déc.-05, à 22:49, Russell Standish a écrit :

On Mon, Dec 05, 2005 at 03:58:20PM +0100, Bruno Marchal wrote:

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.


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".


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.



Yes - this does make sense. Kripke frames are a good way of explaining
why Bp&Dp captures prob=1 type statements. I'm still not sure "bet
on" is the correct verb though, as in normal life one bets on things with
prob <1 (eg on a horse winning a race). Prob=1 is a "sure bet", but I
can't quite think of an appropriate verb.

Well thanks, and "sure bet" is probably better than my phrasing. It is really the particular case of "probability one". I will surely explain asap why Quantum Logic can be interpreted as the logic of "probability one" in quantum mechanics. This has been single out by Maria Louisa Dalla Chiara, the quantum logician of Florence (Firenze) Italy, but it was the main basic motivation of von Neumann when he opened the field of quantum logic.

Let me give you the main result (by Goldblatt) connecting quantum logic and modal logic. It is a theorem of representation of quantum logic into modal logic. There exist a function R translating quantum logic in modal logic. R can be described recursively in the following way:

The atomic statement p are interpreted by the "quantization(*)" BDp. i.e R(p) = BDp
the negative statement ~A is interpreted by B~R(A), i.e, R(~A) = B~R(A)
conjonction: R(A & B) = R(A) & R(B).

Goldblatt proved that MQL proves A iff the modal logic B proves R(A)
MQL is for Minimal Quantum Logic (the result can be extended to the standard orthomodular quantum logic).

And B is the logic having as 1) axioms: K, T, and LASE (p -> BDp), the "little abstract Schroedinger equation); 2) inferences rules: Modus ponens and Necessitation rule.
(Those who have forget should search LASE in the archive).

Exercise: could someone guess which multiverses (W,R) makes LASE valid (true in all worlds for all "illumination" on the worlds)? Answer: R need just to be symmetrical. See why?

More explanation soon. I intend to answer an off-line mail by Stathis who asks good questions, asap (not today, probably tomorrow). Later I will explain how the translation of the UDA (Universal Dovetailer Argument) in the language of a lobian machine gives rise to LASE for the probability 1, the sure bets.



(*) The term quantization in this setting has been introduced by Rawling and Selesnick in a paper where they modelize a quantum NOT with the modal logic B. Reference and abstract here: http://portal.acm.org/citation.cfm?id=347481.


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