On 9/29/2011 4:03 AM, Bruno Marchal wrote:

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:


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.

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.
Is the property of universality independent of whether or not a machine has a set of properties? What is it that determines the properties of a machine? I need to understand better your definition of the word "machine".

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?

What do you mean by computable models?
Allow me to quote several definitions: "computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. " (from http://en.wikipedia.org/wiki/Computable_function). "a computable model is one whose underlying set is decidable and whose functions and relations are uniformly computable. " (from http://arxiv.org/abs/math/0602483). A computable model, as I understand it, could be considered as a representation of a system or structure whose properties can be determined by some process that can itself be represented as a function from the set of countable numbers to itself. This defintion seeks to abstractly represent the way that we can determine the properties of a physical system X or, equivalently, generate a finite list of operations that will create an instance of X.

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.

OK, but then why to use that fact to criticize Jason's defense of arithmetical truth independent of humans.

I am claiming a distinction between the existence of a structure and the definiteness of its properties. It is my claim that prior to the establishment of whether or not a method of determining or deciding what the properties of a structure or system are, one can only consider the possibility of the structure or system. For example, say some proposition or sentence of a language exists. Does that existence determine the particulars of that proposition or sentence? If it can how so? How do can we claim to be able to decide that P_i is true in the absence of a means to determine or decide what P_i means? How do you know the meaning of these word "Unicorn"? Is the meaning of the word "Unicorn" something that that arises simply from the existence of sequence of symbols? is not meaning not something like a map between some set of properties instantiated entity and some set of instances of those properties in other entities? Consider an entity X that had a set of properties x_i that could not be related to those of any other entity? Would this prevent the existence of X?
The existence of X is the necessary possibility of X, []<>X.

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.

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 space, for example.

This paper http://www.jstor.org/pss/2274172 ? It seems to me that the distributivity axiom would not make the same distinction either, although Hilbert space is defined in terms of a linear algebra on a vector space. Consider this paper <http://www.google.com/url?sa=t&rct=j&q=orthomodularity%20axiom&source=web&cd=5&sqi=2&ved=0CDgQFjAE&url=http%3A%2F%2Fm3k.grad.hr%2Fpapers-ps-pdf%2Fquantum-logic%2F1998-helv-phys-acta.pdf&ei=N3SETo2yFYGztwfLiYU0&usg=AFQjCNHal3UDb6B-MATSt1hloWFhSNVCnw&sig2=Fl7ESJLpFZ9qj8c8YU8S-w&cad=rja>'s abstract.


"We show that binary orthologic becomes either quantum or classical logic when nothing but modus ponens rule is added to it, depending on the kind of the operation of implication used. We also show that in the usual approach the rule characterizes neither quantum nor classical logic. The diff erence turns out to stem from the chosen valuation on a model of a logic. Thus algebraic mappings of axioms of standard quantum logics would fail to yield an orthomodular lattice if a unary - as opposed to binary - valuation were used. Instead, non-orthomodular nontrivial varieties of orthologic are obtained. We also discuss the computational efficiency of the binary quantum logic and stress its importance for quantum computation
and related algorithms."

How can we even consider the distinction of one form of abstract structure, such as logical algebras or lattices, from another without there existing a means to generate instantiations of the two? This question goes to the heart of my skepticism of your result.

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.
But do we not decide whether or not to pursue a conjecture by the implications of the conjecture? The questions that I am asking here are questions of the ability of the idea to give us an explanatory narrative that we can use to reason about our world. You are, with your result, proposing a result that implies an ontological theory: that Reality is, at its primitive level, purely abstract. This seems to be more of an echo of the ideas of Pythagoras than those of Plato...

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.

Perhaps. Cerrtainly open problem in comp+Theaetetus.
Does that not imply that the explanatory value of comp+Theaetus is partly dependent of the resolution of such a problem? If we are going to seriously consider your form of ideal monism to be correct, as opposed to some form of non-substance dualism or material monism or neutral monism, do such questions not need to be looked at with seriousness? I am very interested in ontological theories, thus my queries.

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.

I am problem driven. I start from the problem, and use the available math.

As am I. ;-) But I think that sometimes we need to look beyond the math and consider how it is that knowledge itself is possible.

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?

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.

But it seems that you are assuming that our ability to have intuitions of abstractions itself has a satisfactory explanation. You seem to assume that the properties of, for example, memory obtain solely from the existence of Arithmetic and that such existence is severable from the physical instantiations of memory.

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).

How is a representation of logic A in logic B not equivalent to an embedding of A in B? Maybe I am conflating a model with a representation.

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.

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.
A mathematical object, as I can understand it, is purely an abstraction that supervenes upon the actions of a mind to have a meaning. The particular properties of the object flow from the rules, axioms, etc. that are used to define said object and do not depend on anything else except the possibility of some instantiation of those rules, axioms, etc. If a "machine" is a form of mathematical structure then its existence is not predicated on any particular instantiation of such a machine but its properties are not defined by the mere possibility of its existence. Additionally, the notion of "stopping" or "not stopping" has a meaning that refers to a process in some way. A process cannot be reduced to a static relation between abstract entities but it can be represented by sequences of static relations. I distinguish between the representation of a process and the process itself. A map is not the territory. OTOH, if we consider the idea that we can relate simulations of a given process with the process itself, we are comparing one form of process to another, not a static set of relations to a process. I do not think of mathematical objects as static relations only, I see them more as invariant patterns that occur in a background of eternal interactions between possible aspects of Existence.

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.

If we are some abstract static relational structure then Arithmetic is an explanation of everything? Maybe for an abstract and static entity, but not for an entity that needs to explain the appearance of a universe that is never only identically itself. I do not identify an arbitrary collection of static relations with Change in a decidable one to one and onto way.



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