Date: May 18, 2013 7:39 PM
Author: fom
Subject: Re: A logically motivated theory

On 5/18/2013 2:43 PM, Zuhair wrote:
> On May 18, 8:58 pm, fom <> wrote:
>> On 5/18/2013 10:40 AM, Zuhair wrote:

>>> The whole motivation beyond this theory is to extend any first order
>>> predicate by objects.

>> Could you please clarify this remark?

> I'll expand on that. The motivation of this theory is to 'define'
> objects that uniquely corresponds to first order logic predicates,i.e
> for each first order predicates P,Q there exist objects P* and Q* such
> that P*=Q* iff for all x. P(x)<->Q(x)
> We want to do that for EVERY first order predicate P.
> The plan is to stipulate the existence of multiple PRIMITIVE binary
> extensional relations, each one of those would play the role of a set
> membership relation after which object representative of predicates
> are defined.
> Now the first axiom ensures that no 'distinct' objects can be defined
> after equivalent predicates all across the membership relations, so
> although we have many membership relation but yet any objects X,Y that
> are co-extensional over relations E,D (i.e. for all z. z E X iff z D
> Y) are identical!
> The second axiom (schema of course) ensures that no object represent
> non-equivalent Predicates, and so although we do have 'multiple'
> membership relations (primitive extensional relations) however from
> axiom schemas 1 and 2 this would ensure that each object defined after
> any of those relations would stand 'uniquely' for a single predicate.
> The last axiom scheme is just a statement ensuring the existence of an
> object that extends each first order predicate after some membership
> relation.
> Those objects uniquely corresponding for first order predicates are to
> be called as: Sets.
> The point is that paradoxes are eliminated because of having
> 'multiple' extensional relations each standing as a membership
> relation.
> I'm not sure if that would interpret PA, but if it does, then PA can
> be said to be a PURELY logical theory!
> Now if (and this is a big if) we allow infinitely long formulas to
> define first order predicates (infinitary first order languages) then
> second order arithmetic 'might' follow. And I think if this is the
> case, then second order arithmetic is also PURELY logical!
> This mean that the bulk of traditional mathematics (most of which can
> be formulated within proper subsets of second order arithmetic) is
> purely logical!
> However I don't think that higher mathematics can have a pure logical
> motivation comparable to the above, the motivation behind those can be
> said to be 'structural', or 'constructive' in the ideal sense that
> I've presented in my latest philosophical notes on my website and to
> this Usenet.
> Zuhair

Once again, I think you should take a look at NF by Quine.

Note that NF supports Fregean number classes.

You might find the following link informative:

The entire summary at that page is very well written.