
Re: LOGIC & MATHEMATICS
Posted:
May 30, 2013 9:35 PM


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 

