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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 6:15 AM
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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}

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