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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