
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: > > > > > > > > > > > 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 don't see any problem between infinity and logic, also I do maintain that logic with _infinite formulas_ *is* indeed logic, anyhow.
Zuhair

