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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 28, 2013 8:06 AM

On May 28, 7:44 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 26/05/2013 10:17 PM, zuhair wrote:
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> > 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 don't
see any problem between infinity and logic, also I do maintain that
logic with _infinite formulas_ *is* indeed logic, anyhow.

Zuhair

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