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Topic:
A logically motivated theory
Replies:
15
Last Post:
May 21, 2013 8:22 AM




Re: A logically motivated theory
Posted:
May 20, 2013 3:37 PM


On May 20, 2:59 am, fom <fomJ...@nyms.net> wrote: > On 5/18/2013 11:19 PM, Zuhair wrote: > > > > > > > > > > > On May 19, 1:47 am, fom <fomJ...@nyms.net> wrote: > >> On 5/18/2013 2:52 PM, Zuhair wrote: > > >>> On May 18, 10:38 pm, fom <fomJ...@nyms.net> wrote: > >>>> On 5/18/2013 2:21 PM, Zuhair wrote: > > >>>>> On May 18, 8:58 pm, fom <fomJ...@nyms.net> wrote: > >>>>>> On 5/18/2013 10:40 AM, Zuhair wrote: > > >>>>>>> In this theory Sets are nothing but object extensions of some > >>>>>>> predicate. This theory propose that for every first order predicate > >>>>>>> there is an object extending it defined after some extensional > >>>>>>> relation. > > >>>>>>> This goes in the following manner: > > >>>>>>> Define: E is extensional iff for all x,y: (for all z. z E x iff z E y) > >>>>>>> > x=y > > >>>>>>> where E is a primitive binary relation symbol. > > >>>>>> So, > > >>>>>> <X,E> > > >>>>>> is a model of the axiom of extensionality. > > >>>>>>> Now sets are defined as > > >>>>>>> x is a set iff Exist E,P: E is extensional & for all y. y E x <> P(y) > > >>>>>> So, > > >>>>>> xEX <> ... > > >>>>>> where > > >>>>>> ... is a statement quantifying over relations and predicates. > > >>>>> No ... is a statement quantifying over objects. > > >>>> How so? The formula seems to have an > >>>> existential quantifier applying to a > >>>> relation and a subformula with the > >>>> quantified 'E' as a free variable: > > >>>> 'E is extensional' > > >>>> Using 'R' for "Relation", I read > > >>>> Ax(Set(x) <> EREP(extensional(R) /\ Ay(yRx <> P(y)))) > > >>> I meant that P must be first order. There is no General so to say > >>> membership relation E, there are separate different membership > >>> relations all of which are 'primitive' relations. > > >>> Of course one might contemplate something like the following: > > >>> Define(E): x E X iff Exist R Exist P( extensional(R) /\ Ay(yRX < > >>>> P(y) /\ P(x) ) > > >>> This E relation would be something like a 'general' membership > >>> relation, but this is not acceptable here, because it is 'defined' > >>> membership relation and not 'primitive'. When I'm speaking about > >>> membership relations in the axioms I'm speaking about ones represented > >>> by 'primitive' symbols and not definable ones. > > >>> Zuhair > > >> I will refrain from making a long posting based on > >> my earlier mistaken impressions. > > >> However, your remarks here suggest that you should take > >> a look at Quine's "New Foundations" and the interpretation > >> of stratified formulas. If you have Quine's book "Set Theory > >> and Its Logic" available to you, a couple of hours reading the > >> appropriate chapters and flipping forward to the definitions > >> in earlier chapters should give you some sense of the matter > >> as he saw it. > > >> I believe it is Thomas Forster who is making his book > >> available online concerning NF, if you should become more > >> interested in Quine's theory. > > > Yes I'm familiar with NF, actually I managed to further simplify it. I > > coined the Acyclicity criterion, after which we can forsake > > stratification altogether. See the joint article of Bowler, Randall > > Holmes and myself on that subject, you can find it on Randal Holmes > > home page and also on my website. Actually see: > >http://math.boisestate.edu/~holmes/acyclic_abstract_final_revision.pdf > > > Anyhow here I'm trying to achieve something else, that of seeing that > > PA can be interpreted in a LOGICAL theory. I view all the extensional > > primitive relations in this theory as logical since all what they > > function is to extend predicates. If we regard the second order > > quantifier as logical, then that's it the major bulk of traditional > > mathematics belongs to logic. I'm not sure if we can get without the > > second order quantifier. > > > Anyhow I'm not sure of the remarks I've presented here, I might be > > well mistaken, but matters look to go along that side. I'm just > > conjecturing here. > > > DIVIDE and CONCUR > > > Zuhair > > What exactly do you take to be "logical"? > > For example, on the Fregean view, one is interpreting > the syllogistic hierarchy as extensional. This is a > typical mathematical view. However, Leibniz interprets > the syllogistic hierarchy as intensional, and, Lesniewski's > criticisms of Frege and Russell also lead to an intensional > interpretation. > > What you are referring to as "secondorder" is, for me, > a directional issue (extensional=bottomup, intensional=topdown) > with respect to priority in the syllogistic hierarchy. > > One often takes such questions for granted because > our textbooks provide such little background information. > We focus our attentions according to what we are taught. > > John MacFarlane has written on this demarcation issue: > > http://plato.stanford.edu/entries/logicalconstants/ > > Historically, Aristotle is "intensional". This follows > from his claim that genera are prior to species. But, > the problems arise with the issue of "substance". The > notion of "substance" is associated with individuals and > grounds the syllogistic hierarchy predicatively. > > So, there is an implicit tension in foundational studies > because of "firstorder"/"secondorder" or > "extensional"/"intensional" dichotomies. > > That is why I am asking for some clarification as to > what you take to be "logical". > > Thanks.
I don't have a principled approach as regards demarcation of logic yet. For now I'm content with saying that all fragments of first order logic with identity (including all substitution instances by concrete objects) are logical. I also maintain that having object extensions of first order predicates is by itself logical since it just copies the predicative content into the object world. A simple trial to do that is to add a monadic symbol like "e" to the above language and stipulate that
if G is a predicate symbol then eG is a term.
eG is read as "the extension of G".
stipulate the axiom:
eF=eG iff (for all x. F(x) <> G(x))
To me this approach is perfectly logical.
We can use second order quantification to define a membership relation:
x E y iff Exist G: G(x) & y=eG
where G ranges over first order predicate symbols.
In general we can define an i_th membership relation as:
x Ei y iff Exist Gi: Gi(x) & y=eGi
where Gi ranges over the i_th order predicate symbols.
So the membership relations so defined (in second order) only reflect fulfillment of predicates after their order.
Of course to justify such an approach one must show that fulfillment of predicates differs after their order, which indeed is hard to show. Since it seems that "is" in "Socrates is a man", is not really different from "is" in "Triangle is a shape". Of course "is" here is just another word for "fulfills being", so the sentences, completely interpreted, are: "Socrates fulfills being a man", "Triangle fulfills being a shape". Even more completely displayed those sentence are:
The object the name "Socrates" refers to is a man.
The predicate the name "Triangle" refers to is a shape.
It appears that the article "is" in both of the above sentences has the same meaning, that of "fulfills being". And it seems that there is no difference in this fulfillment per se between the two sentences. However still it can be argued that fulfillment of predicates by predicates is a different kind of concept from fulfillment of predicates by objects, and that this difference is the same for higher predicates fulfillment. And this can be a strong point since using extensions in the same manner (that of concatenating the symbol e with the predicate symbol) doesn't elucidate the difference between an object and a predicate and between a predicate and a higher predicate, which are agreeably must be Mirrored by different "sorts" of extensions, so in absence of that difference we must show it by the membership relation. Anyhow, the above stipulation of ordered membership does in sense MIRROR the order of predicates, so in principle it is inert and doesn't add something that is substantially extralogical, so it can be considered as logical. However saying its logical really depends on whether the second order quantifier is inert or not.
Regards
Zuhair



