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 havedifficulty in understanding why Pp=Bp & -B-p should be translatedintoEnglish 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 wellcapturedby Bp&Dp (in world alpha). This means (Kripke-semantically) "p istruein 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. Bruno http://iridia.ulb.ac.be/~marchal/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 "beton" is the correct verb though, as in normal life one bets on thingswithprob <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.`

Cheers, Bruno

`(*) 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.`

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