On 28 Sep 2011, at 16:44, Stephen P. King wrote:
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.
This follows from the UD Argument, in the digital mechanist theory. No
need of AUDA or complex math to understand the necessity of this, once
we accept that we can survive with (physical, material) digital
machines.
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?
What do you mean by 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.
OK, but then why to use that fact to criticize Jason's defense of
arithmetical truth independent of humans.
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, Calude and
von Benthem, and others on this. I am not sure of the definition of
"weak quantum logic" as you use it here.
Svozil, Calude and van Benthem thought on the subject are very good.
Weak quantum logic is the logic of sublattice of ortholattices, like
in the paper of Goldblatt that I have often refer to you. Basically it
is quantum logic without the orthomodularity axiom. It does not
distinguish finite dimensional pre-Hilbert space from Hilbert spave,
for example.
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?
Hard and premature questions.
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 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 , 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.
Perhaps. Cerrtainly open problem in comp+Theaetetus.
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.
I am problem driven. I start from the problem, and use the available
math.
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?
No. Simplicity. Together with the need of the classical Church thesis,
and our intuition of numbers. We do use the comp hypothesis, and it
needs classical logic on the natural numbers. Intuitionist logic can
also be used, but then the math are much more complex, and eventually
we need a non trivial use of the double negation topology. It is more
easy to use, like usually in math, the meta-classical background.
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!
You repeatedly confuse the notion of embedding of a logic in another,
and representing a logic in another. I have explained this many times,
but you keep coming back on that confusion. QL cannot be faithfully
extended in Boolean logic, but this does not mean that you cannot
represent QL in a classical frame work (like it is done all the time;
quantum mechanics is itself a classical theory).
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.
Classical logic allows non constructive reasoning which are obligatory
in any modest theology, like the machine's theologies.
Do you agree that a (mathematical) machine stop or ... do not stop, on
some input. We don't need more than that.
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.
It follows from the step 8 of UDA that if we are machine, classical
arithmetic is a theory of everything. Non classical logics are
recovered in the machine's epistemologies. S4grz1 is intuitionist and
the Z1* and X1* logics are type of quantum logics.
Bruno
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
