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

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fom

Posts: 1,969
Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: May 31, 2013 11:00 PM
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On 5/31/2013 12:45 PM, Zuhair wrote:
> On May 31, 4:35 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>> 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:
>>
>>>> 0=e(contradictory)
>>
>> 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)

>>
>>>> Where 'contradictory' is defined as: for all x. contradictory(x) iff
>>>> ~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
>> ----------------------------------------------------

>
> The post is not about FOL, it is about a kind of predicative
> extensional logic which is second order, all of my comments are about
> that kind of logic and not about FOL. FOL is logic yes but it doesn't
> encompass all kinds of logic, the predicative extensional second order
> logic that I presented in this head post is something that I maintain
> as LOGIC. You and others may disagree, that's fine, but my argument is
> that as far as this system is considered as logic then infinity and
> second order arithmetic follows, and thus promoting logicism in this
> particular sense.
>
> And by the way there is no consensus among mathematicians and
> logicians and philosophers reached yet about what constitutes a
> logical symbol and a logical system, this is a debatable issue. I for
> instance hold that identity, the extension symbol "e" of Frege's, and
> the axiom about it, and the predicative system that I outlined here
> are all logical and purely so, and it is in this sense that I'm saying
> that most of traditional math is reducible to logic since second order
> arithmetic is interpretable in this predicative extensional second
> order logic. However others would even disagree about identity being a
> logical symbol, even some might disagree that classical logic is logic
> and only consider constructive intuitionistic logic as being logic, on
> the other hand some accept first order logic with infinitely long
> formulas as being among logic, others even accept 'tense' symbols
> among logical symbols, etc... a widely debatable issue. I personally
> also accept first order logic with identity and its axioms and
> whatever added primitives predicates and functions without any
> axiomatization about those added primitives to be also a kind of
> *PURE* logic, since the added primitives convey no extra meaning to
> the system other than that inferred from the logical axioms, so in
> this case they are just dummy symbols syntactically extending the
> logical language without having any effect on the logical flaw of that
> system nor on the semantics of it.
>


Nice defense, Zuhair.

I have read Frege and appreciate his "Foundations of Arithmetic".

I have only scratched the surface of Quine's "New Foundations". But,
the fact that it supports Fregean number classes is significant.
Perhaps I will find the time to look at it closely one day.

Your statements make clear that logic is a wonderful and diverse
subject.








Date Subject Author
5/26/13
Read LOGIC & MATHEMATICS
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