On 23 Oct 2013, at 22:57, Russell Standish wrote:

On Wed, Oct 23, 2013 at 03:02:55PM +0200, Bruno Marchal wrote:

On 22 Oct 2013, at 22:50, Russell Standish wrote:

On Tue, Oct 22, 2013 at 03:09:03PM +0200, Bruno Marchal wrote:

On 21 Oct 2013, at 23:03, Russell Standish wrote:


In fact "p-> []p" characterizes sigma_1 completeness (by a
result by
Albert Visser), and that is why to get the proba on the UD*, we use
the intensional nuance []p & <>t  (= proba) starting from G
extended
with the axiom "p-> []p" (limiting the proposition to the UD).


proba?


Sorry - I was actually asking what you meant by the word "proba".

OK. Sorry. It was an abbreviation for probability.


[]p & <>t doesn't seem like a probability.

It is a "probability 1".
von Neumann once argued, if you have a good quantum logic (which means automatically a good axiomatic for the "probability 1" (the box in a modal axiomatic) and the "~probability = 0", (the diamond in the modal approach), all the other probabilities should be derivable from it. Assuming three dimension, and the Hilbert space structure for the quantum state, Gleason theorem get the probabilities from the logic.

I thought some years that I could derive a Temperley Lieb algebra of projection operators in the logic Z1*, but the math get too much complex for me. That would have provided the 3 dimensions (by relation between Temperley-Lieb Algebra and 3D knots), and the Hilbert Space structure, and in that case, the rest is done by Gleason theorem.



Did you mean certainty?

In our context "certainty" might be a bit fuzzy. But I am not against calling a "probability one" a certainty, in case it is an ideal relative certainty, based on the assumption of comp for example, the correct choice of the substitution level, the perfect ability of the doctor/teleportation-machine, all the default hypotheses.


IIRC, one of your hypostases was interpreted as probability=1 (ie
certain) events.

The key is more that []p is not a probability. And that we get an abstract probability (a modal logic of probability or credibility) when we add the consistency (semantically = the existence of at least one accessible reality) condition. So yes, []p & <>t is an arithmetical predicate which "behaves" like a probability one.



Also, is []p & <>t == []p & <>p ?

Yes. Those are definition made in G (and thus in arithmetic), which is a normal modal logic.

<>t implies the existence of an accessible world (by Kripke semantics). []p implies p is true in all accessible world. So there will a world, and p is true in it, so we have <>p when we have []p & <>t . And, of course the reverse is more easy. if we have <>p, we have an accessible world (the one with p true in that world), and t is true there too, as t is true in all world.

Careful! the new logic obtained (with the new box defined by []p & <>t), we lost the necessitation rule, and in that logic, there is no more a Kripke semantics. But we have still a notion of worlds- neighborhoods, which fit better the apparition of topologies, continuua, ... and thus with the UDA way to derive physics.

Bruno










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Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
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