
Re: LOGIC & MATHEMATICS
Posted:
May 31, 2013 1:45 PM


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.
Zuhair

