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:


For well-defined propositions regarding the numbers I think the values are confined to true or false.


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.

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.

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.
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.

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).

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.
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.
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.

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.





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