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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 18 Jan 2011 17:58:56 -0600
Message-id: <0D1107A2-1F8C-4C6A-B897-6EB6AF383D7C@xxxxxxxx>
On Jan 18, 2011, at 4:36 PM, Tara Athan wrote:
In mathematical set theory, a class is a collection of sets but is not itself a set.

That's not quite what the article says.  In  Von Neumann-Bernays-Gödel set theory (VNBG) — the theory on which the entry is based — it is true that all classes are collections of sets, but the same is true of sets, as every set is a class.  Sets are simply those classes that are members of some other class.  Those classes that are not sets — and, hence, not members of any class — are known as proper classes.

Note also that whether there is a set/class distinction depends on the theory. In Zermelo-Fraenkel set theory, there is no such distinction; everything is a set.

This is how I've always interpreted "class" as used in OWL, but I can't speak for other users.

This is not a matter that is open to interpretation (and your interpretation is incorrect).  In the semantics of OWL DL, the extension of a OWL class consists of OWL individuals; in the semantics of OWL Full, there are OWL classes whose extensions include other OWL classes.

So a class has an extension in a particular situation, and that extension is a set, but the extension (of the same class) can be a different set in a different situation. The class is the collection of all of its extensions, unified by its definition/description.

The semantics of OWL per se does not accommodate the idea of a class's extension changing over time, although one could presumably capture the idea formally by means of a series of interpretations (thought of as temporally ordered) that assign different extensions to the same class.  (This is possible because classes are not defined to be identical to their extensions in the semantics.)

Note that it is a recipe for confusion to suppose that the properties of classes according to some mathematical theory of classes transfer unproblematically over to OWL (or any other representation language). If you want to know what a primitive term means according to a language, the only reliable guide is the model theory of the language. Of course, this assumes that you are dealing with a language that, like OWL or any Common Logic dialect, is sufficiently well-defined that it has a model theory. If it doesn't, or if it does but it is simply ignored, then the "semantics" for the language consists of little more than a welter of subjective preferences and vague intuitions.  The prospects for genuine, shareable knowledge representation in terms of such a language are dim at best.

Chris Menzel

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