
Re: LOGIC & MATHEMATICS
Posted:
May 31, 2013 6:24 AM


Nam Nguyen 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?
Different authors define logical symbols in different ways, Zuhair is at liberty to define them as he does.
 I think I am an Elephant, Behind another Elephant Behind /another/ Elephant who isn't really there.... A.A. Milne

