On 28 Sep 2011, at 16:44, Stephen P. King wrote:

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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 thevalues are confined to true or false.Jason --[SPK]Not in general, unless one is only going to allow only Booleanlogics to exist. There have been proven to exist logics that havetruth 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 andits negation. A machine is consistent if it does not assert aproposition and its negation.[SPK]Is not a machine represented mathematically by some abstract(mathematical ) structure? I am attempting to find clarity in theideas surrounding the notion of "machine" and how you arrive at theidea that the abstract notion of implementation is sufficient toderive 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 theoremwhich shows that a theory is consistent if and only if there is amathematical structure (called model) satisfying (in a sense whichcan be made precise) the proposition proved in the theory.[SPK]What constraints are defined on the models by the Gödel-Henkincompleteness theorem? How do we separate out effective consistentfirst-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 onlylogic. There are infinitely many logics, below and above classicalpropositional logic. But this cannot be used to criticize the useof classical logic in some domain.[SPK]OK. My thought here was to show that classical (Boolean) logicis not unique and should not be taken as absolute. To do so would bea 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 muchmore rarely intuitionistic) logic at the meta-level. You will notfind a book on fuzzy logic having fuzzy theorems, for example. Nonclassical logics have multiple use, which are not related with thekind of ontic truth we are looking for when searching a TOE.[SPK]Of course fuzzy logic does not have fuzzy theorem, that could bemistaking the meaning of the word "fuzzy" with the meaning of theword "ambiguous". I have been trying to establish the validity ofthe idea that it is the rules (given as axioms, etc) that are usedto define a given mathematical structure, be it a model, or analgebra, etc. But I think that one must be careful that the logicalstructure that one uses of a means to define ontic truths is notassumed to be absolute unless very strong reasons can be proven toexist for such assumptions.Usually non classical logic have epistemic or pragmatic classicalinterpretations, or even classical formulation, like the classicalmodal logic S4 which can emulate intuitionistic logic, or theBrouwersche modal logic B, which can emulate weak quantum logic.This corresponds to the fact that intuitionist logic might modelizeconstructive provability, and quantum logic modelizesobservability, and not the usual notion of classical truth (as usedalmost everywhere in mathematics).[SPK]I use the orthocomplete lattices as a representation of quantumlogic. My ideas are influenced by the work of Svozil, Calude andvon 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 consideringonly finite emulations of a quantum logic (such as how a classicalapproximation of a QM system could be considered), how might oneapply the Tychonoff, Heine–Borel definition or Bolzano–Weierstrasscriterion of compactness to be sure that compactness obtain for themodels? If we use these compactness criteria, is it necessary thatthe collection of open sets that is used in complete in an absolutesense? COuld it be that we have a way to recover the appearence ofthe axiom of choice or the ultrafilter lemma?

Hard and premature questions.

Could it be possible to have a notion of accessibility toparametrize or weaking the word "every" as in the sentence: " Apoint x in X is a limit point of S if every open set containing xcontains at least one point of S different from x itself." to "Apoint x in X is a limit point of S if every open set , that isassessible from some S, containing x contains at least one point ofS different from x itself. The idea is that S and x cannot be aninfinite distance (or infinite disjoint sequence of open sets) apart.It seems to me that this would limit the implied omniscience ofthe compactness criteria (via the usual axiom of choice) and itseems more consistent with the notion that an emulation does notneed to be *exact* to be informative.

Perhaps. Cerrtainly open problem in comp+Theaetetus.

To invoke the existence of non classical logic to throw a doubtabout the universal truth of elementary statements in well defineddomain, like arithmetic, would lead to complete relativism, giventhat you can always build some ad hoc logic/theory proving thenegation of any statement, and this would make the notion of truthproblematic. The contrary is true.[SPK]Relativism of that kind would be that last conclusion that Iwould desire! OTOH, we do need a clear notion of contextuality asillustrated by the way that words are defined in relation to otherwords 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 aninterpretation of it in the classical framework.{SPK]This seems to be an unnecessary prejudice! Why is the classicalframework 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 believethat QM is derivative of classical logic somehow in spite of myrepeated statements to the work of others that show that this issimply 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 atopological space, provided a classical sense for intuitionistlogic, like a lattice of linear subspaces can provide a classicalinterpretation of quantum logic (indeed quantum logic is born fromsuch structures). It might be that nature observables obeys quantumlogic, but quantum physicists talk and reason in classical logic,and use classical mathematical tools to describe the non classicalbehavior of matter.[SPK]I agree but will point out that the use of classical logic couldbe 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 asfundamental, but this reasoning is build on the intuition that a 3p"public" notion of communication can only be defined in Booleanlogical terms; in other words, we observe a classical realitybecause that is the manner that maximally consistent collections ofopen sets can bisimulate each other. Bisimulation is communicationbetween and within logical systems. If bisimulation cannot occurbetween a pair of logics then there is no interactions between thetopological spaces dual to those logics. This gives us a way tothink of seperate physical worlds. But this reasoning requires thatwe treat logics and topological spaces on an equal ontologicalfooting. Logic cannot be taken as the unique ontological aspect ofexistence.

`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.