On Jun 5, 6:52 pm, Zuhair <zaljo...@gmail.com> wrote: > I want here to elaborate on the relationship between mathematics and > logic. > > First I 'll start explaining what I mean by logic. > > Various methodologies have been proposed to demarcate logic in a > complete manner, i.e. given any concept P we can by using any of those > demarcating methodologies decide upon whether P is logic or not. This > is a principled approach. > > However here I will not adopt such a strong claim, what I'll do is to > start with a system that is fairly accepted as being Logical that no > demarcation method would be accepted if it designates it as non > logical. So I'll rely on a system that our naive understanding > strongly hold as being about logic, so we are starting from a point > that is prior to any demarcation issue. > > Now lets assume that L is such a system, i.e. naively understood as > being "purely logical". The next step is to decide on what kinds of > adjustments involved in EXTENDING L into system L2 would be considered > as *inert*, i.e. no involving an extra-logical concept, as to consider > the resulting system L2 logical! in other words we'll consider > criteria of extending a logical system into a logical system. > > The followings are some: > >  If L is extended by adding whatever amount of primitive symbols > and no additional axiomatization or modification of formation of > formulas or inference rules are stipulated, then this mere addition > will result in a logical system also. > >  If L is extended with a symbol that is justified of being > considered as "logical", and axiomatization about it are also > considered as "logical", then the resulting system is Logical. > >  If L is extended without adding any new primitive symbol, nor any > new axiomatization or inference rule, and the extension is only > effected by stipulating new rules of formation of formulas that are > already been used in accepted purely logical systems, and can be > naively understood as simple technical > syntactical adjustments just required for extending the system in a > consistent manner, then the resulting system is logical. > > If we can lay down specific formal criteria explaining all of those, > then we'd have a hierarchy of logical systems of different powers, > this will enable us to reach about very powerful logical systems in > which most if not all mathematics can be interpreted. > > Now I'll supply an example to show what I mean. > > First we hold that Classical First order logic and all its known > fragments are "pure logical systems". > > Now we may add whatever number of primitives to FOL, if non of those > primitives is axiomatized and no formation or inference rule is > changed, then the "FOL+empty primitives" is what results and is a > "purely logical system". > > Now we add a logical symbol that is Identity "=" and add first order > identity theory axioms "ID", the resulting system "FOL+empty > primitives + = + ID" is also a "purely logical system". > > Of course there is a long debate of whether identity is considered as > a logical concept or not, here I'll side with those accepting it as a > logical concept. > > Now we add the logical monadic symbol e, denoting "extension of", and > axiomatize that if P is a predicate symbol then eP is a term of the > language, and axiomatize in a Fregean manner: > > eP=eQ iff (For all x. P(x) iff Q(x)) .... > > Now we consider that as a logical axiom, the underlying idea is that > the question of "extending" predicates is a purely logical question, > and it is not specific to any kind of first order predicates so it > doesn't depend on the content (semantics) that predicate stands for, > and thus I accept it as a kind of logic that I call as "Extensional > logic", also the idea is that the above axiom is "illustrative" it > illustrates what is going on with predicates, so the additional power > that the system has comes from this illustrative concept, and > illustrating a logical > concept is a logical concept also. So the system "FOL+empty predicates > +=+ID+e+" is a pure logical system. > > Now we desire to upgrade this system into second order realm, so we > need to add variable symbols that range over predicates, impose > necessary restrictions on them, and make adjustment on formation of > formulas necessary to ensure that no paradox will be reached by those > formation rules. > > Contemplate the following extension: > > We divide objects into two sorts, those symbolized in lower case, and > those symbolized in primed lower case. So objects the variable x range > over are not necessarily the same ones variable x' range over. > > We restrict predicate symbols to take object arguments only. > > Predicates to be symbolized in upper case. > > Also predicate symbols shall be divided into two sorts: non primed > predicate symbols and primed predicate symbols. > > We restrict the primed predicate symbols to be only Constant predicate > symbols, so P' actually hold for a particular predicate of the second > sort only. While non primed predicate symbols can be Variables ranging > over predicates of the first sort or might be constant symbols ranging > over one predicate of the first sort. To differentiate those we put > two brackets around the symbol to show that it is constant, this can > also be done with the object symbols. So [P] denotes a specific > predicate of the first sort, while P is a variable ranging over first > sort of predicates. > > Of course since all primed predicates are constants we'll leave it > written without brackets since it cannot be a variable symbol, so P' > is a Constant predicate of the second sort. > > phi, pi, $ are used as specified. > > Now we come to define "well sorted" formula: > > x=y ; x'=y' ; [x]=[y] ; [x]'=[y]' ; x=[y] ; x'=[y]' are well sorted > formulas. > > P'(x') ; P'([x]') are well sorted formulas. > > P(x) ; P([x]) ; [P](x) ; [P]([x]) are well sorted formulas. > > y=e[P] ; y'=eP' are well sorted formulas. > > y'=eP is a well sorted formula. >
Just a further detail, it is worth noting that it is more plausible to "define" constants after variables by using a "witness" concept.
So we can add the symbol w which is a dyadic symbol linking Predicate variable symbol to a formula. The rule is:
if phi is a well sorted formula in which only x1...xn occur free, then wP(for all x1..xn (P(x1..xn) iff phi)) is a term.
This is abbreviated as wP(P:=phi)
":=" stands for "is definable after"
Axiom: wP(P:=phi) := phi
This says that wP(phi) is definable after phi, as P is definable after phi, since wP(P:=phi) is just a predicate that witnesses P, it is nothing but a particular substitute of P, to enact that particularity we state:
so y=e[P] would be enacted after "variable symbols" as
y=e(wP(P:=phi)) for any well sorted formula phi satisifying the conditions above.
This facilitates the proof of second order arithemtic, and make the above method more coherent.
It is noted that this method can be extended over all natural sorts by natural indexing, thus possibley making this logic interpreting 'every' n-th order arithemtic, thus most of mathematics.
> If phi and pi are any well sorted formulas and "--" is a logical > connective, then phi--pi is a well sorted formula. > > If phi is a well sorted formula in which the variable symbol (whether > predicate or object) $ is free, then Exist $. phi is a well sorted > formula. > > Now modify the law about e to the following: > > If phi is a predicate symbol occurring in a well sorted formula then > ephi is a term. > > And Keep axiom  unmodified as above. > > Keep all other axioms unmodified. > > Now all of the above adjustments are mere technical adjustments to > ensure extending the pure logical theory "FOL+empty primitives + = +ID > +e+" in a consistent manner into the second order realm, the > adjustments doesn't > rely on any particular mathematical concept, it is just a pure > technical syntactical modification on formulas as to get a consistent > extension, which employs methodology already present in well known > logical systems like restricting predicates to constant predicate > symbols (FOL), taking only objects as arguments (FOL), the priming is > a simple syntactical technique to avoid paradoxes that involves no > mathematical insights at all, it is used in multi-sorted first order > logical languages. So the goal is "extending a logical system in a > consistent manner" which is a pure logical motivation, and the tools > "formation rules of formulas by multi-sorting and restrictions on > predicates" are used in well known purely logical systems, so they are > logical also, since the motivation and > the tools are logical, then the resulting system is LOGICAL and purely > so. > > The resulting logical system I call it as: Bi-sorted Extensional SOL. > > The main idea is that all work with the above systems only concerns > itself with logic, no external concepts are negotiated at all. > > It is nice to see that the above logical system that mends itself with > logic only do interpret a mathematical theory like Peano Arithmetic! > > So instead of adding to first order logic with identity some five > extra-logical primitives (0,number, successor, +, x), and infinitely > many extra-logical axioms about them, everything can be done in a > purely logical theory upgrading first order logic with identity by > handful amount of adjustments on formation rules. > > The nice thing is that if we upgrade the above system into three sorts > (like using starred, primed, non primed symbols) following the same > lines above, the resulting "purely logical system" I call as "Tri- > sorted Extensional SOL" will actually interpret second order > arithmetic! > > We can use the natural indexing machinery (commonly used in known > pure > logical systems) to actually have all natural sorts in this way I > think every n_th order arithmetic would be interpreted in it. And the > system is clearly purely logical also, since logicality would be > transmitted from the lower sort to the next above. > > Mathematics unlike physics don't require the particular meaning > attached to their symbols even if they are discovered using that > meaning. Since logical theories shares the same feature, then > mathematics would be totally absorbed in logic once it is > interpretable in it, this is unlike physics, empirical, ethics, > linguistics, morals, etc.. theories which extends logic, those would > only be so called if the symbols in the theory hold for the particular > semantics assigned to them, so they cannot be absorbed into logic, > since axioms must be added to characterize that meaning which cannot > be derived by purely reasoning about logical formula formation or the > alike logical issues. > > So it seems 'philosophically speaking' that mathematics is just > symbolic logic really, but the question is: > > is that reduction heuristic? > > I think it should have some heuristic value, but certainly not to > advocate completely vanquishing all of mathematical discovering > endeavor into that about logic. I don't think this would be > heuristically justified, possibly it is justified for some areas of > mathematics but I think not for all, I still think that humans better > make their discoveries in strong ambiance, then afterwards they go > reflect back upon their discoveries and check whether they can be done > in weaker grounds, however I still maintain that such reductions are > not without benefit. And that the > logical hierarchy is useful, and it would add to mathematical > discovery and philosophy as well. > > Zuhair