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

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fom

Posts: 1,968
Registered: 12/4/12
Re: A logically motivated theory
Posted: May 18, 2013 6:47 PM
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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.





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