Mike Carroll
I've been led by a long and circuitous route through logic, topology,
and set theory, to a nonstandard logic. For now, I'm calling this
YANL, for "yet another nonstandard logic", as there are too many

YANL shares at least one feature with systems that have been developed
as "quantum logics": YANL algebras are not boolean algebras. YANL was
not intended as a quantum logic. But that doesn't mean it isn't one.

Does anyone know if anyone has attempted to state criteria of adequacy
for a quantum logic? I suspect not. But I would appreciate any
recommendations anyone can make along these lines, to save me from
sifting through the quantum logic literature from von Neumann to the
present.

Statements of the form "If L is a quantum logic then ...", which allow
one to show that L is *not* a quantum logic, would be especially
welcome.

Thanks for any help.
Charles Francis
A quantum logic has the precisely defined mathematical structure of a
Hilbert space, and it should be very straightforward to check if YANL
has such a structure. See Karl Svozil Quantum logic. A brief outline, in
which there are plenty of other refs.

http://arxiv.org/PS_cache/quant-ph/pdf/9902/9902042.pdf
--
Charles Francis
Mike Carroll
Thank you very much for providing a reference which is both recent and
available online.

The paper, together with some further reflection, has helped me
understand what bothers me about quantum logic as currently
understood.

"The starting point [of quantum logic] is von Neumann's Hilbert space
formalism of quantum mechanics", the paper says on page 1. We then use
this formalism to provide an interpretation of logical operations (pp.
2-3). After examining the resulting logic, we find that "The
propositional system obtained is _not_ a classical Boolean
algebra...." (p.5).

What bothers me is that a Hilbert space is a topology, constructed
using set theory, which in turn relies upon a logic whose
propositional system is in fact a classical boolean algebra. Why not
replace classical boolean algebra to begin with, rather than accept it
first and then replace or reject it later upon discovering that it
does not meet our needs for physics?

("Easier said than done," might be one reply.)

The moderator, however, may feel I am straying too far afield. Thanks
in any case for the reference and its excellent bibliography.

Mike Carroll
Oro Valley, AZ
Arnold Neumaier
This is because quantum logic is clumsy to use and does not help to solve
any significant quantum application. Almost everything done to apply
quantum mechanics is based on classical logic for the Schroedinger equation.

Charles Francis
Another might be that it does not meet the needs of mathematics. In
mathematics we seek the simplest consistent axiom set, and reduce
everything to that. In mathematics we want to be able to show that a
many valued logic is a consistent mathematical structure by reducing it
to Boolean logic. In physics we only want to use it when appropriate,
having been assured that it is consistent to do so.

Another is that it does a different job from Boolean algebra. In physics
we can understand definite statements

"the particle is measured at position x", which has a Boolean truth
value provided we have actually done a measurement

Bayesian statements, with probabilistic truth values

"If I do a measurement I will measure the particle at x"

Much less understood are hypothetical statements

"If I were to do a measurement I would measure the particle at x"

which we apply to the case where we know perfectly well that we are not
going to do a measurement (e.g. asking which slit did the particle come
through in a Young's slit experiment)

I see quantum logic as giving truth values for these hypothetical, or
counterfactual statements (in Boolean logic you always have p->q is true
when p is false, which does not fit an intuitive notion there is some
truth in describing what would happen in other circumstances.

I don't know that physics requires any other form of statement, or any
other logic other than Boolean, Probability theory and quantum logic.
And although I do see some value in Fuzzy Logic, I think what is being
described there is psychology rather than physics.

>The moderator, however, may feel I am straying too far afield.

I would think not. Foundations of quantum theory and the application of
mathematics to physics are both very much on topic.

--
Charles Francis
QUOTE
Does anyone know if anyone has attempted to state criteria of adequacy for a quantum logic?

1) The orthomodular lattice of projection in a complex Hilbert space
should
satisfy the axioms.

2) The "logic" should be "enough" states (i.e. measures)

3) It should be "useful", that is it should allow us to "compute"
or to "understand" something new.

4) It is useful to have some kind of "Noether's theorem" holding, that
would guarantee that there is a relation between one parameter groups of
authomorphisms and "conserved quantities", at least for some
"representations" (or "presentations" or "realizations") of you logic

That is about all what can be said in general. The rest of the devil is
in the details.

ark
Charles Francis
With the notable exception of understanding
--
Charles Francis
Mike Carroll
QUOTE
... In mathematics we seek the simplest consistent axiom set, and reduce everything to that. In mathematics we want to be able to show that a many valued logic is a consistent mathematical structure by reducing it to Boolean logic. In physics we only want to use it when appropriate, having been assured that it is consistent to do so.

This gives Boolean logic an authority, independent of physics, which
it does not merit. We invented Boolean logic, and we may decide that
it is too simple. One of the best kinds of evidence for its being too
simple is that we have trouble mapping quantum logic to Boolean logic.

QUOTE (->
 QUOTE ... In mathematics we seek the simplest consistent axiom set, and reduce everything to that. In mathematics we want to be able to show that a many valued logic is a consistent mathematical structure by reducing it to Boolean logic. In physics we only want to use it when appropriate, having been assured that it is consistent to do so.

This gives Boolean logic an authority, independent of physics, which
it does not merit. We invented Boolean logic, and we may decide that
it is too simple. One of the best kinds of evidence for its being too
simple is that we have trouble mapping quantum logic to Boolean logic.

I see quantum logic as giving truth values for these hypothetical, or counterfactual statements (in Boolean logic you always have p->q is true when p is false, which does not fit an intuitive notion there is some truth in describing what would happen in other circumstances.

This is presented as a difference between Boolean and quantum logic. I
would take it instead as a criticism of Boolean logic, that it fails
to provide a logic for counterfactual statements. Many logicians would
agree that this is a problem with Boolean logic, apart from any

How could Boolean logic be "too simple"? It is possible to add a
closure operator to the language of Boolean algebra. We can then add
postulates for the closure operator. Linguistically, this adds to
Boolean logic some of the expressive power of topological spaces. This
has already been investigated to some extent: Sikorski, "Boolean
Algegras", section 41, "Topology in Boolean algebras. Applications to
non-classical logic."

The "non-classical" logic that Sikorski refers to are modal logics,
which are also the logics of hypotheticals and counterfactuals. This
may be a coincidence but I doubt it.

To summarize, it seems to me that the current state of quantum logic
indicates that Boolean logic is in need of revision. I will however
have to learn considerably more than what I now know about Hilbert
spaces, before I can see whether there is any truth to my conjecture.

(A longer version of these comments was posted earier, but seems to
have disappeared.)
Jjacksonjackson
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