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

 Messages: [ Previous | Next ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: LOGIC & MATHEMATICS
Posted: May 30, 2013 9:35 PM

On 30/05/2013 4:54 AM, Zuhair wrote:
> On May 30, 1:15 pm, Zuhair <zaljo...@gmail.com> wrote:
>> On May 29, 5:29 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>
>>
>>
>>
>>
>>
>>

>>> 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!

>>
>> Infinity: Exist x (0 E x & (for all y. y E x -> {y} E x))
>>
>> where E is defined as in the head post.

FOL doesn't have 'E' (as in your "{y} E x") as a logical symbol.

>>
>> while 0 and {y} are defined as:
>>

That's a bizarre concoction of symbols as far as FOL logical symbols
are concerned: on both sides of '=' there are _invalid_ FOL logical
symbols, namely '0' and 'e'. Iow, '0' and 'e' aren't FOL logical
symbols.

>> {y}=e{isy}
>
> a typo
> correction: {y}=e(isy)

>>
>> ~x=x
>> and 'isy' is defined as: for all x. isy(x) iff x=y
>>
>> I consider the monadic symbol "e" as a "logical" symbol, also identity
>> symbol is logical.
>>
>> The above infinity is a theorem of this logic.

I did specifically specify "FOL" when I posed the challenge. Right?

--
----------------------------------------------------
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