Search All of the Math Forum:

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

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: A logically motivated theory
Posted: May 20, 2013 9:23 PM
 Plain Text Reply

On 5/20/2013 2:37 PM, Zuhair wrote:
> 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 "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.

>
> 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.

This is where your statement

P*=Q* iff P<->Q

gets interesting.

Frege and Russell both pursued description theories. Russellian
description theory actually does not involve reference since it
had been formulated for logical realism (mutually exclusive
truth valuation). It is a type of quantifier that is either
instantiated or not instantiated.

So, when you address the question of "concreteness", many
issues become involved.

I will not pursue this. But as you know, Quine introduced his
set theory with an analysis of how identity becomes eliminable.

This is distinct from set theory as presented in Kunen or
Jech. Both introduce the axiom of extension with a conditional
and defer to the received paradigm of first-order logic for its
converse.

With regard to descriptions -- or, more precisely, denotations --
Zermelo's original paper treated identity in terms of denotations
in relation to singletons. Later developments changed the
description of the domain.

What I find interesting about your statement is that it is
relating a first-order relation (admittedly taken as a "logical"
symbol by some authors) with a zero-order connective.

I have thought about this a great deal in my own deliberations.

> 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 review of Feferman's work suggested by Alan Smaill in
another thread described predicativism as a "framework"
applied to "something given".

Now, in Aristotle, one finds the observation that
interpreting universal quantification as a course-of-values
defeats the intention of the quantifier. It would
seem that empiricism demands this intepretation.

I am interpreting your statement along those lines. In
other words, following Frege one "cannot really say what
an object is". So, your notion of predicativism is taking
"the world" as given. Moreover, it is taking "the world"
pluralistically -- that is, "the world" is not given in
the singular.

Thus extensions are "witnesses". That is,
the objects that satisfy the concepts reflect the
formula,

Ex(phi(x)) -> phi(c)

where c is constant naming an instantiating object.

Would this summary reflect your views?

> 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.
>

This corresponds to the usual interpretation
of Frege's distinction between the concept and
its extension.

> 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.
>

So, you do not use "general quantifiers". In other
words, Your predicativism is much like Russell's and
your multiplicity of extension relations corresponds
with typed quantifiers. Would that be correct?

The Lesniewskian criticism would correspond with
general quantification. Actually, it has to do with
existential import. In predicative logicism, the objects
are existentially prior to the class. For Lesniewski,
the class and its constituents are existentially
simultaneous.

But, Lesniewski did not describe it this way. He used
the extensional/intensional distinction. But, this notion
of predicativism seems more than simple extensionality
because of the typing of quantifiers.

> 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.

This is where Russellian description theory raises its
head. If singular denotation corresponds with a quantifier,
then there is no distinction between fulfillment by objects
and fulfillment by predicates. "Objects" instantiate
descriptions. Descriptions are "concrete" predications. The
hierarchy is a hierarchy of predications.

Maybe I have oversimplified this.

> 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
> extra-logical, so it can be considered as logical. However saying its
> logical really depends on whether the second order quantifier is inert
> or not.

When Brouwer criticized Hilbert's program, he described a
system of "twoness". Although he claimed to be implementing
a notion of Kant's a prioriness of time, the description is
far from Kantian. The sense of what you are describing reminds
me of Brouwer's statements (but not his motivating description).

Indeed, there is an initial "twoness". The posterior form for each
instance of "twoness" becomes the prior form for the next instance
of "twoness".

For Brouwer, this progression is that of "being" and of "coming to
be" in some Hegelian, or even Nietzchean sense. A typed hierarchy
does not reflect this temporal sense. But, its form of existential
priority is "uniform" with this same sense of "twoness". Each
pair of consecutive levels in the hierarchy has this relation. But,
the pairs overlap.

In any case, thank you for your clarifications.

You might find Russell's discussion of set existence and the axiom
of reducibility in the first edition of "Principia Mathematica" of
interest here. When set existence is presumed, the extensionality
relations are predicative. Of course, the axiom of reducibility
had been what Russell perceived as necessary for the "no-classes"
theory of the first edition. But the discussion seems vaguely
related to some of your remarks in this last paragraph. So, it
may be of interest.

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

© The Math Forum at NCTM 1994-2018. All Rights Reserved.