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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