fom
Posts:
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Registered:
12/4/12


Re: LOGIC & MATHEMATICS
Posted:
May 31, 2013 11:00 PM


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 nonlogical 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 _nonlogical_ >>>>>>>>> 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 _nonlogical_ . >> >>>>>>> The reason is quite simple: in the language L of FOL (i.e. there's no >>>>>>> nonlogical 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 nonlogical 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.

