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

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 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: LOGIC & MATHEMATICS
Posted: May 28, 2013 10:29 PM

On 28/05/2013 6:06 AM, Zuhair wrote:
> On May 28, 7:44 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> On 26/05/2013 10:17 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, and then I expand
>>> it by concepts that are very similar to the kind of concepts that made
>>> it, for example here in the above system you only see rules of
>>> formation of formulas derived by concepts of constants, variables,
>>> quantifying, definition, logical connectivity and equivalents,
>>> restriction of predicates. All those are definitely logical concepts,
>>> however what is added is 'extension' which is motivated here by
>>> reduction of the object/predicate/predicate hierarchy, which is a
>>> purely logical motivation, and also extensions by the axiom stated
>>> would only be a copy of logic with identity, so they are so innocuous
>>> as to be considered non logical.
>>> That's why I'm content with that sort of definitional extensional
>>> second order logic as being LOGIC. I can't say the same of Z, or ZF,
>>> or the alike since axioms of those do utilize ideas about structures
>>> present in mathematics, so they are mathematically motivated no doubt.
>>> NF seems to be logically motivated but it use a lot of mathematics to
>>> reach that, also acyclic comprehension uses graphs which is a
>>> mathematical concept. But here the system is very very close to logic
>>> that I virtually cannot say it is non logical. Seeing that second
>>> order arithmetic is interpretable in it is a nice result, it does
>>> impart some flavor of logicism to traditional mathematics, and
>>> possibly motivates logicism for whole of mathematics. Mathematics
>>> might after all be just a kind of Symbolic Logic as Russell said.

>>
>>> Zuhair
>>
>>>>> similarly here
>>>>> although recursion is used yet still we are speaking about logic,
>>>>> formation of formulas in the above manner is purely logically
>>>>> motivated.

>>
>>>> "Purely logically motivated" isn't the same as "purely logical".
>>
>>> A part from recursion, where is the mathematical concept that you
>>> isolate with this system?

>>
>> I don't remember what you'd mean by "this system", but my point would be
>> the following.
>>
>> In FOL as a framework of reasoning, any form of infinity (induction,
>> recursion, infinity) should be considered as _non-logical_ .
>>
>> The reason is quite simple: in the language L of FOL (i.e. there's no
>> non-logical symbol), one can not express infinity: one can express
>> "All", "There exists one" but one simply can't express infinity.
>>
>> Hence _infinity must necessarily be a non-logical concept_ . Hence the
>> concept such the "natural numbers" can not be part of logical reasoning
>> as Godel and others after him have _wrongly believed_ .
>>
>> Because if we do accept infinity as part of a logical reasoning,
>> we may as well accept _infinite formulas_ and in such case it'd
>> no longer be a human kind of reasoning.
>>

>
> I see, you maintain the known prejudice that the infinite is non
> logical? hmmm... anyhow this is just an unbacked statement.

I did; you just don't recognize it apparently: my "The reason is quite
simple:" paragraph.

> I don't see any problem between infinity and logic,

Well, then, why don't you express infinity with purely logical
symbols, for us all in the 2 fora to see? Seriously, that would
be a great achievement!

> also I do maintain that
> logic with _infinite formulas_ *is* indeed logic, anyhow.

Anyway, First Order _Logic_ does _not_ admit _infinite formulas_ !

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

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