On 05 Feb 2014, at 22:24, Russell Standish wrote:
On Wed, Feb 05, 2014 at 02:57:15PM +0100, Bruno Marchal wrote:
On 05 Feb 2014, at 02:37, Russell Standish wrote:
I understand that Bp&Dt gives one of von Neumann's quantum logics,
but
it still seems an enormous jump from there to the FPI,
It will be the other way round. By UDA we have the FPI. To translate
that FPI we need to define "probability or measure on consistent
extensions" in arithmetic. By the FPI, to say that I will necessary
drink coffee after the WM-duplication, means that I will get coffee
in all consistent extensions (here W and M).
I don't see how a probability=1 concept helps with the FPI, which
requires probability < 1 (p=0.5 at a minimum, for two equally probable
possibilities).
P(I get a coffee) = 1, in case you get a coffee in both W and M.
P(laws of physics) = 1 (the laws of physics are invariant for the comp
continuations, as they are all defined globally on all computations).
It is an extreme ideal case, but it makes sense. P = 1 for the simple
transportation, in the theoretical context, with all default
assumptions.
The []p will ensure that p is true in all accessible worlds.
But unfortunately, []p is true for all p in the cul-de-sac world (a
future exercise for Liz!), which shows that provability is not a
probability, nor a measure of certainty. To get the certainty, we
have to explicitly assume at least one accessible world, and this is
done by imposing "<>t", which imposes one accessible world, and
makes disappear the cul-de-sac possible situation.
I appreciate the no cul-de-sac result in the []p&<>p hypostase, but
does that really mean much when Kripke frames are lost? I guess you
replace Kripke by Scott-Montegue, is that right?
Yes, but the notion is still defined in G, where we have the Kripke
structure.
Then we loose the Kripke semantics, but this means only, here, that we
get quasi-filters instead of filters. The semantics has to changed,
but that is rather a good new, as we get some refinement on the notion
of neighbors. Yes, for Z and X, and Z1 and X1, we can adopt the
semantic of Scott-Montague, but we lost them on the [ ]* extensions.
It is replaced by infinite sequences of such structures (which fit
well with the probability on UD*).
or to call the
Deontic relation a Schroedinger equation, even a little abstract
one.
The deontic relation is []p -> <>p.
The "little schroedinger equation" will be
p -> []<>p (together with []p -> p),
Ah - it was 15 years ago when I read your (Lille) thesis
cover-to-cover. My confusion no doubt comes from you introducing both
concepts on the one page. I still slightly get the french words
"encore" and "deja" confused, because they were introduced on the same
page in my French text book.
It is the one bringing back the symmetry, and leading to the quantum
logic, and the proximity spaces (where the measure will live),
thanks to Goldblatt results.
I'd be interested in the "proximity spaces". Is this a new result, or
just some speculation?
It is an old general result. Take an ortho-space, for example the
linear vectorial space, with a scalar product. In such space, you can
look at the lattice of all linear subspaces. you can interpret "&" by
intersection of subspaces (they are subspaces), and you can interpret
the "V", on two subspaces, by the least subspace containing the two
subspaces.
Ah, and you can interpret the negation, of A, by the greater subspace
orthogonal to A.
Then, if you interpret the atomic sentences, p, q, r by rays in that
linear space, you get a (minimal) quantum logic.
The probability to go from "p" to "q" is given, by QM, by the scalar
product of normalized vector (rays) and is the square of the cosine of
the angle between the two rays. OK?
Let us a put a Kripke structure where world are quantum states, and
thus subspaces. we say that p is accessible from g if the scalar
product is not 0 (= if they are not orthogonal). That can be
generalized on the subspaces.
That define a quantum proximity relation ("not being orthogonal"). It
is reflexive and symmetrical.
But in comp we go in the other way: we get the ortho-structure, by the
semantics of p->[]<>p + []p -> p, and we define the proximity relation
by the non-orthogonality.
This is" known" (by quantum logicians). For example the logic B (p-
>[]<>p + []p -> p) is used to define a proximity relation on "vague
predicate" in a study on vagueness by Williamson(*). Then Goldblatt
use it also.
The reverse is exploited too.
Start from a Kripke multiverse M1, obeying p->[]<>p and []p -> p
(making the multiverse accessibility relation R1 symmetrical and
reflexive, which I am explaining to Liz)
Then build a multiverse M2, with the same worlds than M1, but with the
*complementary* accessibility relation R2. That means:
(alpha R1 beta) in M1 if and only if ~(alpha R2 beta) in M2.
The reflexive relation get irreflexive, and the symmetries all
vanishes. In fact you get an ortho-structure, from the proximity
relation.
That is advanced material. We might come back on this; to see why B
axiomatizes simultaneously a quantum logic, and a proximity relation
for rays and subspaces.
The basic idea is that orthogonal = not proximous, and proximous = not
orthogonal (= scalar product non null).
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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