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

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namducnguyen

Posts: 2,688
Registered: 12/13/04
Re: LOGIC & MATHEMATICS
Posted: May 28, 2013 11:02 PM
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On 27/05/2013 10:44 PM, Nam Nguyen 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_ .


By "logical reasoning" I meant "pure logical reasoning"

> 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.


Again, "pure logical reasoning".
>
> In fact in such case we'd consider ourselves as God.
>
> For one, I'd certainly not consider myself so.



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

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


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

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