On 17 Jun 2014, at 19:51, Platonist Guitar Cowboy wrote:
On Tue, Jun 17, 2014 at 7:17 PM, Bruno Marchal <[email protected]>
wrote:
Thanks. It looks interesting. K is amazing by itself. It is "löbian"
in the sense that the theorems of K are closed for the Löb rule: if
K proves []A -> A, for some modal formula A, then K proves A. []
([]A->A)->[]A is true about K.
I will take a look when I have the times, and I hope it is not
"trivial", as K is indeed very weak and very general, and I could
argue that there is some substance (pun) in Birkhoff and von Neumann.
I felt a bit uneasy about this going through the paper with
"refutation" ringing in my head, so any observations are most
welcome :-) PGC
Quantum logic usually designates the logical structure associated to
the lattice of the subspaces of an (infinite dimensional) Hilbert
space, where lives the atomic physical states (the rays, or unit
vectors, the so called pure states). A base of pure states define an
observable, and the linear structure of the Hilbert spaces determine
the yes-no logic obeyed by the observable. Typical axioms of classical
logic are violated, like the distributivity (a & (b V c) is no more
equivalent with (a & b) V (a & c). The logic is rich, but miss the
tensor products to get close to the quantum formalism per se. Also,
von Neumann algebras and non commutative geometry formalism can be
related, although nothing is very easy there. QL can also be related
to quantum computation, but here too, the relation are not trivial at
all.
When I say that comp + classical theory of knowledge is refutable, I
mean that you can compare the QL infered by empiric studies, and the
QL given by Z1*, X1*, S4Grz1.
van Fraassen wrote a paper entitled "the labyrinth of quantum logics",
but comp provides only three one, and it should be compared to the
more reasonable (empirically) quantum logic. the comparison must be
done in term of the "measure one" logic, not necessarily in term of
this or that formalism, which can ofetn be related by representation
theorems.
UDA should explain why we have to proceed in this way, and the
advantage is that we get the nuances, on the physical reality, between
the core physics, the geography, the communicable, the sharable, etc.
The translation in arithmetic is made necessary by the self-reference
incompleteness (Gödel, Löb) and the nuances on provability brought by
that incompleteness.
May be I am quick explaining the importance of the logic of self-
reference, but UDA is based only on self-referential question (like
probability of surviving here or there).
Feel free to ask for any precision. (Just expect some answer delays
due to June business).
Bruno
He might also be fuzzy on observer. The comp hypothesis
automatically enrich the normal and non normal modalities.
Bruno
On 16 Jun 2014, at 08:16, meekerdb wrote:
This may be of interest.
Brent
Quantum Logic as Classical Logic
Simon Kramer
(Submitted on 13 Jun 2014)
We propose a semantic representation of the standard quantum
logic QL within the classical, normal modal logic K via a lattice-
embedding of orthomodular lattices into Boolean algebras with one K-
modal operator. Thus the classical logic K is a completion of the
quantum logic QL. In other words, we refute Birkhoff and von
Neumann's classic thesis that the logic (the formal character) of
Quantum Mechanics would be non-classical as well as Putnam's thesis
that quantum logic (of his kind) would be the correct logic for
propositional inference in general. The propositional logic of
Quantum Mechanics is modal but classical, and the correct logic for
propositional inference need not have an extroverted quantum
character. The normal necessity K-modality (the weakest of all
normal necessity modalities!) suffices to capture the subjectivity
of observation in quantum experiments, and this thanks to its
failure to distribute over classical disjunction. (A fortiori, all
normal necessity modalities that do not distribute over classical
disjunction suffice.) The key to our result is the translation of
quantum negation as classical negation of observability.
Subjects: Quantum Physics (quant-ph); Logic in Computer Science
(cs.LO); Mathematical Physics (math-ph); Logic (math.LO); Quantum
Algebra (math.QA)
Cite as: arXiv:1406.3526 [quant-ph]
(or arXiv:1406.3526v1 [quant-ph] for this version)
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