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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 19, 2013 7:59 PM
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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 "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.











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