On 18 Jun 2014, at 22:49, Platonist Guitar Cowboy wrote:
On Wed, Jun 18, 2014 at 8:52 PM, Bruno Marchal <[email protected]>
wrote:
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'm not sure we're on the right level here, as I wasn't precise
enough. Apologies.
My misunderstanding. No problem :)
I meant the paper's claim that Van Neumann thesis is refuted, that
logic of QM is non-classical. I think I can see the outlines of the
point, but my answer would still be "yes and no!' at this point. PGC
Indeed von Neumann theorem, actually, made hidden variable impossible
(and thus classical logic irretrievable), but his theorem was
"refuted" by some counter-examples (to be short). Those were non-
local, and later Bell will show that the (local) non-locality of
quantum mechanics is testable, and has been tested (Aspect, but also
many quantum gates) confirming the non classical logic of the local
observations.
Similar no-go theorem in quantum mechanics have been found like the
Kochen-Specker theorem.
The many-worlds, saves locality and determinacy for the multiverse,
but the appearances of non-locality and indeterminacy are explained by
the relative states of the universal schroedinger wave or heisenberg
matrix.
And then, as you know, my point is that is that if mechanism is true,
the heisenberg matrix can be extracted from a sort of statistics on
the universal numbers' possible "dreams" (computations seen from the
1p). I apply math on the mathematician (the dreamer) like Everett
applied physics on the physicians.
Bruno
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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