On 9/27/2011 10:47 AM, Bruno Marchal wrote:
On 27 Sep 2011, at 13:49, Stephen P. King wrote:
On 9/26/2011 7:56 PM, Jason Resch wrote:
<snip>
For well-defined propositions regarding the numbers I think the
values are confined to true or false.
Jason
--
[SPK]
Not in general, unless one is only going to allow only Boolean
logics to exist. There have been proven to exist logics that have
truth values that range over any set of numbers, not just {0,1}.
Recall the requirement for a mathematical structure to exist:
Self-consistency.
Consistency is a notion applied usually to theories, or (chatty)
machines, not to mathematical structures.
A theory is consistent if it does not prove some proposition and its
negation. A machine is consistent if it does not assert a proposition
and its negation.
[SPK]
Is not a machine represented mathematically by some abstract
(mathematical ) structure? I am attempting to find clarity in the ideas
surrounding the notion of "machine" and how you arrive at the idea that
the abstract notion of implementation is sufficient to derive the
physical notion of implementation.
In first order logic we have Gödel-Henkin completeness theorem which
shows that a theory is consistent if and only if there is a
mathematical structure (called model) satisfying (in a sense which can
be made precise) the proposition proved in the theory.
[SPK]
What constraints are defined on the models by the Gödel-Henkin
completeness theorem? How do we separate out effective consistent
first-order theories that do not have computable models?
Also, it is true that classical (Boolean) logic are not the only
logic. There are infinitely many logics, below and above classical
propositional logic. But this cannot be used to criticize the use of
classical logic in some domain.
[SPK]
OK. My thought here was to show that classical (Boolean) logic is
not unique and should not be taken as absolute. To do so would be a
mistake similar to Kant's claim that Euclidean logic was absolute.
All treatises on any non classical logic used classical (or much more
rarely intuitionistic) logic at the meta-level. You will not find a
book on fuzzy logic having fuzzy theorems, for example. Non classical
logics have multiple use, which are not related with the kind of ontic
truth we are looking for when searching a TOE.
[SPK]
Of course fuzzy logic does not have fuzzy theorem, that could be
mistaking the meaning of the word "fuzzy" with the meaning of the word
"ambiguous". I have been trying to establish the validity of the idea
that it is the rules (given as axioms, etc) that are used to define a
given mathematical structure, be it a model, or an algebra, etc. But I
think that one must be careful that the logical structure that one uses
of a means to define ontic truths is not assumed to be absolute unless
very strong reasons can be proven to exist for such assumptions.
Usually non classical logic have epistemic or pragmatic classical
interpretations, or even classical formulation, like the classical
modal logic S4 which can emulate intuitionistic logic, or the
Brouwersche modal logic B, which can emulate weak quantum logic. This
corresponds to the fact that intuitionist logic might modelize
constructive provability, and quantum logic modelizes observability,
and not the usual notion of classical truth (as used almost everywhere
in mathematics).
[SPK]
I use the orthocomplete lattices as a representation of quantum
logic. My ideas are influenced by the work of Svozil
<http://tph.tuwien.ac.at/%7Esvozil/publ/publ.html>, Calude
<http://www.cs.auckland.ac.nz/%7Ecristian/10773_2006_9296_OnlinePDF.pdf> and
von Benthem <http://staff.science.uva.nl/%7Ejohan/publications.html>,
and others on this. I am not sure of the definition of "weak quantum
logic" as you use it here.
One question regarding the emulations. If one where considering
only finite emulations of a quantum logic (such as how a classical
approximation of a QM system could be considered), how might one apply
the Tychonoff, Heine--Borel definition or Bolzano--Weierstrass criterion
of compactness to be sure that compactness obtain for the models? If we
use these compactness criteria, is it necessary that the collection of
open sets that is used in complete in an absolute sense? COuld it be
that we have a way to recover the appearence of the axiom of choice or
the ultrafilter lemma?
Could it be possible to have a notion of accessibility to
parametrize or weaking the word "every" as in the sentence: " A point
/x/ in /X/ is a *limit point* of /S/ if every open set
<http://en.wikipedia.org/wiki/Open_set> containing /x/ contains at least
one point of /S/ different from /x/ itself." to "A point /x/ in /X/ is a
*limit point* of /S/ if every open set
<http://en.wikipedia.org/wiki/Open_set> , that is assessible from some
S, containing /x/ contains at least one point of /S/ different from /x/
itself. The idea is that S and x cannot be an infinite distance (or
infinite disjoint sequence of open sets) apart.
It seems to me that this would limit the implied omniscience of the
compactness criteria (via the usual axiom of choice) and it seems more
consistent with the notion that an emulation does not need to be *exact*
to be informative.
To invoke the existence of non classical logic to throw a doubt about
the universal truth of elementary statements in well defined domain,
like arithmetic, would lead to complete relativism, given that you can
always build some ad hoc logic/theory proving the negation of any
statement, and this would make the notion of truth problematic. The
contrary is true.
[SPK]
Relativism of that kind would be that last conclusion that I would
desire! OTOH, we do need a clear notion of contextuality as illustrated
by the way that words are defined in relation to other words in a
dictionary.
A non classical logic is eventually accepted when we can find an
interpretation of it in the classical framework.
{SPK]
This seems to be an unnecessary prejudice! Why is the classical
framework presumed to be the absolute measure of acceptability and, by
implication, Reality? This statement seems to reveal an explanation of
why you believe that QM is derivative of classical logic somehow in
spite of my repeated statements to the work of others that show that
this is simply not possible except in a crude and non-faithful manner!
A non standard truth set, like the collection of open subsets of a
topological space, provided a classical sense for intuitionist logic,
like a lattice of linear subspaces can provide a classical
interpretation of quantum logic (indeed quantum logic is born from
such structures). It might be that nature observables obeys quantum
logic, but quantum physicists talk and reason in classical logic, and
use classical mathematical tools to describe the non classical
behavior of matter.
[SPK]
I agree but will point out that the use of classical logic could be
merely a habit and convenience. I think that there may be a reason why
classical logics are taken as fundamental, but this reasoning is build
on the intuition that a 3p "public" notion of communication can only be
defined in Boolean logical terms; in other words, we observe a classical
reality because that is the manner that maximally consistent collections
of open sets can bisimulate each other. Bisimulation is communication
between and within logical systems. If bisimulation cannot occur between
a pair of logics then there is no interactions between the topological
spaces dual to those logics. This gives us a way to think of seperate
physical worlds. But this reasoning requires that we treat logics and
topological spaces on an equal ontological footing. Logic cannot be
taken as the unique ontological aspect of existence.
Onward!
Stephen
Bruno
http://iridia.ulb.ac.be/~marchal/
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