Re: [ontolog-forum] Non-Well Founded Sets

[Adrian Walker <adriandwalker@xxxxxxxxx>]

Avril, Mathew --

Something similar comes up with the "Barber" version of Russell's paradox.

You can state the paradox in Executable English [1] like this

some-person lives in the village
not : that-person shaves himself
-----------------------------------------
B shaves that-person


this-person shaves this-other-person
=============================


this-person lives in the village
=======================
 A
 B
 C


this-person is a barber
==================
 B

However, the reasoning system [2] does a syntax check and objects that the above would be meaningless if executed.

You can try this out by pointing a browser to the same site and selecting the example called RussellParadox.

                                     -- Adrian

[1]  www.reengineeringllc.com/A_Wiki_for_Business_Rules_in_Open_Vocabulary_Executable_English.pdf

[2]  Internet Business Logic
A Wiki and SOA Endpoint for Executable Open Vocabulary English Q/A over SQL and RDF
Online at www.reengineeringllc.com    
Shared use is free, and there are no advertisements

Adrian Walker
Reengineering

On Sat, Dec 18, 2010 at 6:28 AM, Avril Styrman <Avril.Styrman@xxxxxxxxxxx> wrote:

Dear all,

I'm not aware of anything that requires to be modeled by non-well
founded sets.


In some computer applications, self-reference appears to work without
difficulties. We can have e.g. a picture of Earth taken from space. We
zoom into some continent, country, city, house, through the window of
the house, and in the room we have a picture of Earth taken from
space. We zoom into that image, and we get to the start of the process
again: we have a picture of
Earth taken from space. We can zoom boundlessly many times. Now, is
this a case of self-reference? —no, it was only a computer system that
connected together a variety of photos, a system that only appeared to
implement self-reference, but it did not.

http://www.ted.com/talks/blaise_aguera_y_arcas_demos_photosynth.html

Non-well founded sets can be used to model this. Imagine that there
are three zoom-levels: A) a picture of Earth taken from space; B) a
picture of a country; C) a picture of a room where is the picture A
among other things. Now, we have the sets A={B}, B={C}, and C={A},
that can be used to model the situation. We can zoom from A to B, from
B to C, and from C back to A. But why should we use non-well founded
sets to model this? It would be simpler just to say that there is a
link from C to A; when you klick C, the link takes you to A. This is
what is done in practice; there is no need for a non-well founded
foundation for a thing that does not require it.


-Avril

Lainaus "Matthew West" <dr.matthew.west@xxxxxxxxx>:

> Dear Chris M and Chris P,
>
> Can I suggest we take this to the Ontolog Forum, I don't think it is really
> of interest for this year's ontology summit, and I will respond there.
>
> Regards
>
> Matthew West
> Information  Junction
> Tel: +44 560 302 3685
> Mobile: +44 750 3385279
> matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
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>
>
>> -----Original Message-----
>> From: ontology-summit-bounces@xxxxxxxxxxxxxxxx [mailto:ontology-summit-
>> bounces@xxxxxxxxxxxxxxxx] On Behalf Of Chris Partridge
>> Sent: 16 December 2010 22:37
>> To: 'Ontology Summit 2011 discussion'
>> Subject: Re: [ontology-summit] FW: [ontolog-invitation] Invitation to a
>> brainstorming call for the 2011 Ontology Summit
>>
>> Not every logician agrees with Chris (which I am sure is no surprise to
>> anyone - including Chris).
>>
>> For example,
>> http://www.amazon.co.uk/gp/product/1575860082/ref=wms_ohs_product
>> Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena - Jon
>> Barwise (Author), Lawrence Moss (Author)
>>
>> Product Description
>> Circular analyses of philosophical, linguistic, or computational phenomena
>> have been attacked on the assumption that they conflict with mathematical
>> rigour. Barwise and Moss have undertaken to prove this assumption false.
>> This volume is concerned with extending the modelling capabilities of set
>> theory to provide a uniform treatment of circular phenomena.
>>
>> Jon Barwise (http://en.wikipedia.org/wiki/Jon_Barwise ) argues that the
>> circularity found in non-well-foundedness is common in the real world -
> i.e.
>> it is a common requirement. As such it would make sense to include it on
>> your foundation.
>> While personally not finding all his examples persuasive, I think the
>> general point that it is a requirement is well made.
>>
>> Another interesting recent (technical) book on a similar the same topic is
>> http://www.amazon.co.uk/gp/product/0199276439/ref=wms_ohs_product -
> Absolute
>> Generality - Agustín Rayo (Editor), Gabriel Uzquiano (Editor).
>> I believe the introduction can be found somewhere on the net.
>>
>> Chris
>>
>> > -----Original Message-----
>> > From: ontology-summit-bounces@xxxxxxxxxxxxxxxx [mailto:ontology-
>> > summit-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
>> > Sent: 16 December 2010 20:53
>> > To: Ontology Summit 2011 discussion
>> > Subject: Re: [ontology-summit] FW: [ontolog-invitation] Invitation to a
>> > brainstorming call for the 2011 Ontology Summit
>> >
>> > On Dec 16, 2010, at 2:24 PM, Matthew West wrote:
>> > >> -1 for 15926, with arguments:
>> > >> http://ontology.buffalo.edu/bfo/west.pdf
>> > >
>> > > Which are answered in:
>> > > http://www.matthew-
>> > west.org.uk/documents/Reponse%20to%20Barry%20Smith%20Comments%2
>> > 0on%20ISO%2015926.htm
>> >
>> > Barry's criticisms of the use of a non-well-founded set theory like
>> Aczel's AFA
>> > are on the money. He notes that it is a greatly overpowered for the
> needs
>> of
>> > the document; it entails, among other things, the entire massively
>> infinite
>> > hierarchy of  transfinite numbers.  (How massive?  So massive that there
>> is
>> > no transfinite number big enough to number them.)  Moreover, ironically,
>> > AFA and its like are in a sense underpowered as well for the given task.
>> > Notably, as I understand the document, THING is itself a class that
>> contains,
>> > well, everything.  The existence of such a class (understood as a non-wf
>> set)
>> > is flatly inconsistent with non-wf ZF spinoffs like AFA.
>> >
>> > Bottom line (as John Sowa likes to say): The underlying class theory of
>> the
>> > document needs to be thrown out and rethought completely.
>> >
>> > Chris Menzel
>> >
>> >
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Always forward towards the supreme maxim of scientific philosophizing

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