On 21 Feb 2012, at 17:53, meekerdb wrote:
On 2/21/2012 7:38 AM, Bruno Marchal wrote:
Negative amplitude of probability comes from the formula p->[]<>p
satisfied by the sigma_1 arithmetical sentences (that is the UD).
How does that work?
By using a theorem of Goldblatt which shows that:
MQL proves A iff B proves t(A),
MQL being a Minimal form of Quantum Logic and B being the Brouwersche
modal logic, with axioms
[ ](A->B) -> ([ ]A -> [ ]B), [ ]A -> A, A->[ ]<>A, and the Modus
Ponens rule + the necessitation rule.
and the translation t(A) given by
t(p) = [ ]<> p p atomic
t(A & B) = t(A) & t(B)
t(~A) = [ ] ~t(A)
So B models an quantum orthologic, a bit like S4, or S4Grz, are known
to model Intuitionist Logic.
Now, the UD is modeled by the restriction of the arithmetical
realisation of modal logic to the Sigma_1 arithmetical sentences, for
which it can be shown that [ ](p->q) -> ([ ]p -> [ ]q), [ ]p -> p, p-
>[ ]<>p, that is the axioms of B, when [ ]p is the result of the
material hypostases translation, which I sum up often by Bp & Dt (but
which is really given by a translation like above).
The MP rule is sound, but we lose the necessitation rule. Nevertheless
we obtain still a quantum logic by using the "reverse-Goldblatt
translation", leading to an arithmetical interpretation of a sort of
quantum logic, and this where the UDA shows we need to find the
elementary logic of the yes/no observable in the comp physical reality.
That was the first step in the verification that the comp physics fit
empirical physics. Others have followed (orthomodularity, a violation
of a Bell type of inequality). Unfortunately the translation of those
nested modal operators (with many "[ ]<>") makes the algorith
intractable for more interesting "physical formula".
Hard to explain this without being technical, but more is said in
sane04 and some other papers in my url.
Bruno
http://iridia.ulb.ac.be/~marchal/
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