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To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Chris Partridge" <mail@xxxxxxxxxxxxxxxxxx>
Date: Wed, 21 Jan 2009 14:08:13 -0800 (PST)
Message-id: <20090121220813.F2321138CE8@xxxxxxxxxxxxxxxxx>
In-Reply-To: <78A1B804-C432-44CA-AC48-54BB1056683D@xxxxxxx>
Subject: RE: [ontolog-forum] Next steps in using ontologies as standards
Date: Wed, 21 Jan 2009 22:12:54 -0000
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Hi Pat,    (02)

Thanks for this.    (03)

A few more clarification questions - and a request for explanation.    (04)

> > PH>when some basic advances in logic showed that the traditional
> > 'layering' of descriptions into individuals/classes/properties/
> > metaclasses/etc. was (a) not necessary and (b) expressively very
> > restrictive. One can keep the categories but abandon the strict
> > layering - in effect, allowing a given thing to be in many 'layers'
> at
> > once - and no disasters arise, if one cleaves to a certain simple,
> > natural syntactic discipline (which is built into both Common Logic
> > and RDF). The result is greatly increased expressivity and a
> formalism
> > which 'naive' users invariably find quite natural, and which makes
> > perfect semantic sense.
> >
> > Is there somewhere we can find more details on this 'basic advance'?
> It was the development of non-well-founded set theory, written up in a
> Stanford CSLI monograph by Peter Aczel. There are articles on it on
> Wikipedia and other places, for a full account see
> http://plato.stanford.edu/entries/nonwellfounded-set-theory/
> . The key point is that set theories which do not use, and even which
> explicitly deny, the axiom of foundation, are not only possible, but
> can be proven to be relatively consistent with ZFC. So they are just
> as 'good' as a foundation as anything else. So, there is absolutely no
> reason to prohibit self-containing sets such as rdfs:Class. If you
> look at classical FOL in this light, you quickly end up with Common
> Logic.    (05)

I thought you may be talking about Aczel (wrt rdfs:Class)- I think if =
looks you can find a pdf of his paper (Matthew and I, at least, have
downloaded it). BTW I recall Aczel keeping extension, but he defined it =
in a
different way - but it quite a while since I read him. I also liked =
& Moss's 1996, Vicious Circles, for its descriptions of the motivations.    (06)

Aczel gives us self-membered classes. But I am not sure how this gives =
the rest of the de-layering you mentioned
(individuals/classes/properties/metaclasses/etc.). In old-fashioned set
theory (under a common interpretation (Wiener)), relations are just sets
with a particular internal structure - so the this is close to a
de-layering.    (07)

I can see how your layering comments apply to FOL (rather than set =
So should I understand things as follows?    (08)

Individuals - what the quantifiers range over
Classes - unary predicates
Properties - non-unary predicates
Metaclasses - predicates that take predicate class predicates=20    (09)

> > For example, can you clarify what is meant by 'individual' here?
> I use this word strictly in the logical sense, to refer to any entity
> in the universe of discourse, any thing in the set that the
> quantifiers are understood to range over. The word in this sense is
> most emphatically not a classifier word. Any kind of thing can be an
> individual in this sense.=20
> > I know
> > there are a range of possible senses. I assume that here it not
> > individual
> > in the Aristotelian sense of primary substance
> Indeed not.  That notion does not even make sense, IMO. =20    (010)

I am afraid I could not resist asking why you say this does not make =
If you mean the Aristotelian notions of primary and secondary substance =
I can see your point of view. However, if you mean the distinction =
particularity and generality, then I am not so sure. Surely it is =
(though maybe na=EFve) to think what distinguishes a particular, such as =
Hayes, from a class of things, such as the class of Welshmen, is that =
Hayes cannot have members, whereas the class of Welshmen can and does.    (011)

Chris    (012)

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