Hi Bruno, ok, I have not yet had the time to study modal logic (it is on my list, but intermediate future). Thanks for the Goldblatt reference.
The paper is not online, but I found it in this book which is at our University Library, maybe interesting also for other people: Goldblatt, Mathematics of Modality http://www.amazon.com/Mathematics-Modality-Center-Language-Information/dp/1881526240/ref=sr_1_1?ie=UTF8&s=books&qid=1232402154&sr=8-1 (the book contains the full paper) Cheers, Günther Bruno Marchal wrote: > > On 16 Jan 2009, at 22:04, Günther Greindl wrote: > >> Hi all, >> >> the question goes primarily to Bruno but all other input is >> welcome :-)) >> >> Bruno, you said you have already arrived at a quantum logic in your >> technical work? > > > > Yes. The hypostases, with p restrict to the Sigma-1 sentences (the > UD) given by > > Bp & p (the knower certainty) > Bp & Dp (the observer certainty) > Bp & Dp & p (the "feeler" certainty), with B the Godel Beweisbar > predicate, and Da = ~B~a. > > gives rise to Brouwersche like modal logics with natural quantization > (BDp) which act like quantum projector, except that I loose the > Brouwersche necessitation rule, which formally makes things more > complex, more rich also. > > > > >> >> May I refer to the following two paragraphs?: >> >> We can read here: >> http://plato.stanford.edu/entries/qt-quantlog/ >> >> The Reconstruction of QM >> >> From the single premise that the “experimental propositions” >> associated >> with a physical system are encoded by projections in the way indicated >> above, one can reconstruct the rest of the formal apparatus of quantum >> mechanics. The first step, of course, is Gleason's theorem, which >> tells >> us that probability measures on L(H) correspond to density operators. >> There remains to recover, e.g., the representation of “observables” by >> self-adjoint operators, and the dynamics (unitary evolution). The >> former >> can be recovered with the help of the Spectral theorem and the latter >> with the aid of a deep theorem of E. Wigner on the projective >> representation of groups. See also R. Wright [1980]. A detailed >> outline >> of this reconstruction (which involves some distinctly non-trivial >> mathematics) can be found in the book of Varadarajan [1985]. The point >> to bear in mind is that, once the quantum-logical skeleton L(H) is in >> place, the remaining statistical and dynamical apparatus of quantum >> mechanics is essentially fixed. In this sense, then, quantum >> mechanics — >> or, at any rate, its mathematical framework — reduces to quantum logic >> and its attendant probability theory. > > > > Very nice text. I agree, but it is a difficult matter. You can extract > the quantum of 1 bit, but the quibit needs a good tensor product, > which is not easy to derive (unless in ad hoc way) from quantum logic. > With comp, I think we will need the first order extension of the > "hypostases", and it could be that special feature of computability > theory will need to be discovered to complete the derivation. In my > 1991 paper I sum by saying that comp is in search of its Gleason > theorem". A lot of work remains, of course. > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
