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

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: LOGIC & MATHEMATICS
Posted: May 30, 2013 6:54 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.
>
> while 0 and {y} are defined as:
>
> 0=e(contradictory)
> {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.
>
> Zuhair





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
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namducnguyen
5/26/13
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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
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Charlie-Boo
5/26/13
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Zaljohar@gmail.com
5/27/13
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zuhair
5/27/13
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fom
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Zaljohar@gmail.com
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fom
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namducnguyen
5/28/13
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Zaljohar@gmail.com
5/28/13
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namducnguyen
5/29/13
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5/30/13
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namducnguyen
5/30/13
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Peter Percival
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Zaljohar@gmail.com
5/30/13
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Zaljohar@gmail.com
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namducnguyen
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Zaljohar@gmail.com
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Zaljohar@gmail.com

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