On Jan 21, 2009, at 4:08 PM, Chris Partridge wrote: (01)
> mc.us>
> 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,
>
> Thanks for this.
>
> A few more clarification questions - and a request for explanation.
>
>>> 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'?
>> =20
>> 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.
>
> I thought you may be talking about Aczel (wrt rdfs:Class)- I think
> if =
> one
> 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 (02)
Yes. The Menzel trick was at first just a way to model Aczel-ian sets
in a conventional set theory, but the intensionality emerged as a
beneficial side-effect. (03)
> - but it quite a while since I read him. I also liked =
> Barwise
> & Moss's 1996, Vicious Circles, for its descriptions of the
> motivations.
>
> Aczel gives us self-membered classes. But I am not sure how this
> gives =
> us
> the rest of the de-layering you mentioned
> (individuals/classes/properties/metaclasses/etc.). (04)
It doesn't require the full unravelling, but it permits it. And we
had other motivations for wanting to do this, arising from the
intended use of the logic on the Web. Thought to be honest, we (Chris
Menzel and I) began it just to see if we could, purely as an
intellectual exercise. (05)
> 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.
>
> I can see how your layering comments apply to FOL (rather than set =
> theory).=20
> So should I understand things as follows?
>
> Individuals - what the quantifiers range over
> Classes - unary predicates
> Properties - non-unary predicates (06)
Yes so far, but ... (07)
> Metaclasses - predicates that take predicate class predicates=20 (08)
I have never understood what the intended meaning of "metaclass" was.
After learning mathematical set theory, it is hard to imagine what
could be more "meta" than a set. (09)
In CL there really aren't such things as classes and metaclasses,
etc.. There are simply names, which name things. And a thing can be
treated as having any logical category you want it to have. If you
want it to be a relation, it can be. If you want it to be an argument
of a relation, it can be. Hence, all things are (potentially)
relations, and all relations are (potentially) meta-relations. The old
layer classifications cease to have any logical meaning. Of course, if
you want to keep to the layered discipline, the logic does not prevent
you, but it does not impose this upon your axioms as a necessity of
using the logic. (010)
>>> For example, can you clarify what is meant by 'individual' here?
>> =20
>> 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
> ...
>> =20
>>> I know
>>> there are a range of possible senses. I assume that here it not
>>> individual
>>> in the Aristotelian sense of primary substance
>> =20
>> Indeed not. That notion does not even make sense, IMO. =20
>
> I am afraid I could not resist asking why you say this does not make =
> sense.
> If you mean the Aristotelian notions of primary and secondary
> substance =
> then
> I can see your point of view. (011)
Quite. (012)
> However, if you mean the distinction =
> between
> particularity and generality, then I am not so sure. Surely it is =
> natural
> (though maybe na=EFve) to think what distinguishes a particular,
> such as =
> Pat
> Hayes, from a class of things, such as the class of Welshmen, is
> that =
> Pat
> Hayes cannot have members, whereas the class of Welshmen can and does. (013)
It is natural at first blush, but it gets quite hard to keep it up for
an extended length of time. For example, one way to reconcile the
temporal fights is to introduce the notion of a series of time-
snapshots of a 4D entity, this being the nearest thing in the 4D world
to a 3D continuant. If you do that to (the 4-D) Pat Hayes, then I
become the set of my instantaneous snapshots. Now, I am quite happy to
be thought of this way, and it is sometimes very useful. But if we
have a logically rigid distinction between things with members and
things without, then this violates a fundamental partitioning of the
universe. LIke the continuant/occurrent distinction, this dichotomy
seems natural but in fact just gets in the way when one gets down to
serious ontology engineering. One of the great merits of the CL
absolute type freedom is that it imposes no a priori logical obstacles
to such re-conceptualizations of entities. (014)
Pat (015)
>
>
> Regards,
> Chris
>
>
>
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