Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

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

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,454 Registered: 2/15/09
Re: A logically motivated theory
Posted: May 19, 2013 10:41 PM

On May 19, 4:59 pm, 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
> >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 "second-order" is, for me,
> a directional issue (extensional=bottom-up, intensional=top-down)
> 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/logical-constants/
>
> 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 "first-order"/"second-order" or
> "extensional"/"intensional" dichotomies.
>
> That is why I am asking for some clarification as to
> what you take to be "logical".
>
> Thanks.

It seems both a tautology as "coming together" and identity as "being
as one" need to be accommodated, with each as of extensionality (being
equal). For the sweep of tautology from genera combining or as one to
next, then as to the locus of identity of speciation, again the
structures define the meaning of extensionality and its use. Here the
tautology defines equality: given the structure, while identity
defines equality: given the specific in representations.

Then maybe it's a good idea that equality is achievable variously,
here somewhat inspired by Frege's basic systems as explicated in the
article fom references, and then that the methods used to so define
equality for elements of the same system may be interchangeable (self-
evident or series-evident, say) or not, along the lines of
implications of the transfer principle and assigning specific meanings
to for each/ for all / for every, and building into the universal
quantifier, or acknowledging as its construction, general support for
the contrapositive, with regards to existence and uniqueness of items,
in as to differentiating course-of-values and individuated
specification, then of course as relevant to impredicate, yet true,
features of the structure.

Then, as above, Al-Johar, that a particular object is defined by a
first-order predicate, one notion is that some n'th order predicate
(and N'th order), results in the same object, with that the first-
order predicate defines an object, for a representation, while the
n'th-order (as it were) predicate of the structure of the genera,
refers to the same individual, but it is not interchangeable the
evaluation of the predicates that so denote it without having that
acknowledged in the carriage of the structure.

What you describe is by no means irregular, and indeed that's rather
again the point that "pathological" structures like the Universe
_aren't_ regular, yet, their specification is of the most concise (eg,
for any x, x e U with quantification and containment, that a species
and genera of U applies to x with the inward view). Here then the
"outward" view as to the well-founded and constructle and the "inward"
view as of the total and continuous sees that: your system here is
much similar to Frege's: where's the system that accommodates both
representation: and context.

Then, with the goal to be a "logical" logical theory toward structures
of mathematical use: the elements of integer arithmetic have a total
structure, the context, a logical theory for all of these objects has
only logical objects in the theory, then, all the regular logical
all: the theory should be the same from where those are
indistinguishable.

Then, in terms of the arithmetic coding of some dually-self-
infraconsistent ur-object, for that there is a completed infinity: of
the general extension via deduction.

question is: what is the theory of Nothing and Everything at once? I
answer it with the "null axiom theory", a logical theory with no non-
logical axioms, "logical".

Regards,

Ross Finlayson

Date Subject Author
5/18/13 Zaljohar@gmail.com
5/18/13 fom
5/18/13
5/18/13 Zaljohar@gmail.com
5/18/13 fom
5/18/13 Zaljohar@gmail.com
5/18/13 fom
5/19/13 Zaljohar@gmail.com
5/19/13 fom
5/19/13 ross.finlayson@gmail.com
5/20/13 Zaljohar@gmail.com
5/20/13 fom
5/20/13 fom
5/21/13 Zaljohar@gmail.com
5/18/13 Zaljohar@gmail.com
5/18/13 fom