
Re: LOGIC & MATHEMATICS
Posted:
May 30, 2013 6:15 AM


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.
while 0 and {y} are defined as:
0=e(contradictory) {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.
Zuhair

