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Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: May 27, 2013 10:50 AM

On 5/27/2013 4:51 AM, Zuhair wrote:
> On May 27, 7:51 am, fom <fomJ...@nyms.net> wrote:
>> On 5/26/2013 11:17 PM, zuhair wrote:
>>
>>
>>
>>
>>
>>
>>
>>
>>

>>> On May 26, 4:49 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>> On 26/05/2013 3:52 AM, Zuhair wrote:
>>
>>>>> On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>>> On 26/05/2013 12:52 AM, Zuhair wrote:
>>
>>>>>>> Frege wanted to reduce mathematics to Logic by extending predicates by
>>>>>>> objects in a general manner (i.e. every predicate has an object
>>>>>>> extending it).

>>
>>>>>> [...]
>>
>>>>>>> Now the above process will recursively form typed formulas, and typed
>>>>>>> predicates.

>>
>>>>>> Note your "process" and "recursively".
>>
>>>>>>> As if we are playing MUSIC with formulas.
>>
>>>>>>> Now we stipulate the extensional formation rule:
>>
>>>>>>> If Pi is a typed predicate symbol then ePi is a term.
>>
>>>>>>> The idea behind extensions is to code formulas into objects and thus
>>>>>>> reduce the predicate hierarchy into an almost dichotomous one, that of
>>>>>>> objects and predicates holding of objects, thus enabling Rule 6.

>>
>>>>>>> What makes matters enjoying is that the above is a purely logically
>>>>>>> motivated theory, I don't see any clear mathematical concepts involved
>>>>>>> here, we are simply forming formulas in a stepwise manner and even the
>>>>>>> extensional motivation is to ease handling of those formulas.
>>>>>>> A purely logical talk.

>>
>>>>>> Not so. "Recursive process" is a non-logical concept.
>>
>>>>>> Certainly far from being "a purely logical talk".
>>
>>>>> Recursion is applied in first order logic formation of formulas,
>>
>>>> Such application isn't purely logical. Finiteness might be a purely
>>>> logical concept but recursion isn't: it requires a _non-logical_
>>>> concept (that of the natural numbers).

>>
>>>>> and all agrees that first order logic is about logic,
>>
>>>> That doesn't mean much and is an obscured way to differentiate between
>>>> what is of "purely logical" to what isn't.

>>
>>> Yes I do agree that this way is not a principled way of demarcating
>>> logic. I generally tend to think that logic is necessary for analytic
>>> reasoning, i.e. a group of rules that make possible to have an
>>> analytic reasoning. Analytic reasoning refers to inferences made with
>>> the least possible respect to content of statements in which they are
>>> carried, thereby rendering them empirically free. However this is too
>>> deep. Here what I was speaking about do not fall into that kind of
>>> demarcation, so it is vague as you said. I start with something that
>>> is fairly acceptable as being "LOGIC", I accept first order logic
>>> (including recursive machinery forming it) as logic,

>>
>> Compositionality is an intrinsic part of the modern
>> conception of logic. Originally, the idea had been
>> to recognize this in the analysis of natural language,
>> and, this is contemporary in the sense that linguistic
>> analysis is directed at identifying a presumed underlying
>> logical form.
>>
>> This is actually part of the basis for the Frege-Hilbert
>> distinction. Frege viewed logical form as corresponding
>> to underlying thoughts that could be taken as "the same"
>> across differing colloquial language use. Thus, the
>> Fregean notion differed from formalist semantical theories
>> that ground the modern recursive definition of formal
>> languages.
>>
>> Frege's identity puzzles are deprecated in the modern
>> paradigm that treats the axiom
>>
>> Ax(x=x)
>>
>> as an ontological statement of self-identity. But,
>> it is simultaneously a semantical statement of
>> consistent interpretation of syntactically identical
>> inscriptions.
>>
>> The relevance of this is that the denotations of
>> a logical system must be understood as a well-order.
>> If one uses the symbol,
>>
>> 'a'
>>
>> to act as a denoting symbol it can no longer be
>> available as a denoting symbol when the next
>> designation of such is required. So, the
>> course-of-values interpretation of a domain
>> involves the presupposition of a sequential
>> listing of denotations.

>
> Despite the detail of that, the question of whether or not predicates
> have extensions in the Fregean manner is itself a purely logical
> issue. I don't see it a mathematical issue, nor do I see it as a
> mathematical interpretation of logic or anything like that.

the "mathematics of succession" since the nature of
logic in its analytical role arrive at the same structural
form. There had been no intention to dispute your
views.

> So whether
> one stipulate a theory in its favor or against it, then that
> stipulation is a logical answer. And given that identity is itself a
> logical concept,

I disagree with that particular view, but I am not
disputing your use of it as such.

> then there is no harm at all in considering those
> extensions as part of logic as far as they are stipulated in a
> consistent manner, one can view them as an IMAGE of that consistent
> fragment of second order logic with identity on the OBJECT world, so
> it is copying the content of predication in that consistent fragment
> of logic, so it do have a purely logical content, there is no way to
> call that mathematical or any other discipline that logic can be
> applied to. Extensional logic is logic. Now that mathematics is found
> to be interpretable in it, then this yield that mathematics is a part
> of extensional logic. Do you see any clear mathematical motivation
> with the system I've presented?
>

is "yes". But, since I am aware of Frege's work and respect
it profoundly, my qualified answer is "no".

I find it intriguing that Frege's number classes are
representable in Quine's New Foundations. It may be
that a "shadow mathematics" will arise because of the
renewed interest in Frege and the recent serious consideration
of Quine. Only time will tell as these matters are sorted
out further.

Just so you understand some of my "non-standard" commentary,
recall that Frege's "number of a concept" is taken by
analogy from "direction of a line". Along similar lines,
Cantor took the identity of completed infinities in analogy
with the single-point compactification of the plane and
his notion of "equipollence" between powers of an infinite
set generalizes perspectivity in projective geometry.

I understand certain basic relations
geometrically,

news://news.giganews.com:119/Jr2dnbdYvtfPdlrNnZ2dnUVZ_t-dnZ2d@giganews.com

and I understand compositionality through an
intensional axiomatized equational theory,

news://news.giganews.com:119/IqudndogJ8-VB1zNnZ2dnUVZ_qydnZ2d@giganews.com

Truth table semantics begins with geometric
forms instantiating all possibilities for
logical equivalence,

news://news.giganews.com:119/zsCdnW9U7v4BOlzNnZ2dnUVZ_hmdnZ2d@giganews.com

and ends with a canonical ordering that
fixes a representation for a complete
connective,

news://news.giganews.com:119/AuqdnYcXm8eaLVzNnZ2dnUVZ_h-dnZ2d@giganews.com

These particular constructions actually
become quite complex. Certain group theoretic
constructions reflecting the choice of
representation provide a syntactic labeling
of the free orthomodular lattice on two generators.

My understanding of propositions is
based on a free DeMorgan algebra,

news://news.giganews.com:119/Jr2dnbNYvtf9cFrNnZ2dnUVZ_t-dnZ2d@giganews.com

and, in fact, I view DeMorgan algebra
as foundational rather than Boolean
algebra,

news://news.giganews.com:119/Jr2dnbBYvtedcFrNnZ2dnUVZ_t-dnZ2d@giganews.com

The ortholattice O_6 which appears in these
constructions is fundamental to models of
logic as discerned by Pavicic and Megill in

http://arxiv.org/pdf/quant-ph/9906101v3.pdf

Although I have nothing to show for my efforts,
I have taken the issue of demarcation very
seriously. After all, what exactly is meant
by "foundational" if one is starting somewhere
in the middle?

have been intended to argue with your investigation.
As noted before, I am aware of the Fregean point
of view and I respect it. You are doing interesting
things with your analyses, even if I am not quite
fluent enough in the details to always make some

Date Subject Author
5/26/13 Zaljohar@gmail.com
5/26/13 namducnguyen
5/26/13 Zaljohar@gmail.com
5/26/13 namducnguyen
5/26/13 Peter Percival
5/26/13 namducnguyen
5/26/13 Peter Percival
5/26/13 namducnguyen
5/26/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/28/13 Charlie-Boo
5/26/13 Zaljohar@gmail.com
5/27/13 zuhair
5/27/13 fom
5/27/13 Zaljohar@gmail.com
5/27/13 fom
5/28/13 namducnguyen
5/28/13 Zaljohar@gmail.com
5/28/13 namducnguyen
5/29/13 Peter Percival
5/30/13 namducnguyen
5/30/13 Peter Percival
5/30/13 Peter Percival
5/30/13 namducnguyen
5/31/13 Peter Percival
5/30/13 Bill Taylor
5/30/13 Peter Percival
5/30/13 Zaljohar@gmail.com
5/30/13 Zaljohar@gmail.com
5/30/13 namducnguyen
5/31/13 Peter Percival
5/31/13 Zaljohar@gmail.com
5/31/13 LudovicoVan
5/31/13 fom
5/28/13 Peter Percival
5/28/13 namducnguyen
5/27/13 Charlie-Boo
5/27/13 fom
5/28/13 Charlie-Boo
5/28/13 fom
6/4/13 Charlie-Boo
6/4/13 fom
6/5/13 Zaljohar@gmail.com
5/28/13 Zaljohar@gmail.com
5/28/13 LudovicoVan
5/28/13 ross.finlayson@gmail.com
5/28/13 LudovicoVan
5/28/13 LudovicoVan
5/28/13 fom
5/29/13 LudovicoVan
5/29/13 fom
5/30/13 LudovicoVan
5/29/13 fom
5/30/13 LudovicoVan
5/30/13 fom
5/31/13 LudovicoVan
5/31/13 Zaljohar@gmail.com
5/31/13 LudovicoVan
5/31/13 ross.finlayson@gmail.com
6/1/13 LudovicoVan
6/1/13 namducnguyen
6/1/13 ross.finlayson@gmail.com
6/2/13 LudovicoVan
6/2/13 ross.finlayson@gmail.com
6/3/13 Shmuel (Seymour J.) Metz
6/3/13 ross.finlayson@gmail.com
6/4/13 LudovicoVan
6/4/13 namducnguyen
6/4/13 Peter Percival
6/5/13 Shmuel (Seymour J.) Metz
6/5/13 fom
6/6/13 Peter Percival
5/31/13 fom
6/1/13 LudovicoVan
6/1/13 fom
6/2/13 ross.finlayson@gmail.com
6/2/13 fom
6/2/13 Herman Rubin
6/2/13 fom
6/2/13 LudovicoVan
6/3/13 Herman Rubin
6/3/13 Peter Percival
6/4/13 Herman Rubin
6/4/13 Peter Percival
6/4/13 Peter Percival
6/1/13 fom
6/1/13 LudovicoVan
6/1/13 namducnguyen
6/5/13 Peter Percival
6/1/13 fom
6/2/13 LudovicoVan
6/2/13 fom
5/28/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/27/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/30/13 Zaljohar@gmail.com