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.
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