Slight amplification and very minor correction:On Jan 18, 2011, at 5:58 PM, Christopher Menzel wrote: 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.
It might be worth emphasizing that the term 'class' as used in vNBG set theory really has no relation to the sense of the word use in OWL and similar languages. OWL can be completely interpreted within any conventional set theory (eg ZF) without invoking proper classes in the vNBG sense or indeed the set/class distinction at all. The term 'class' in the OWL literature comes from its use as a shorthand for 'classification', and should not be understood as connoting any contrast with the word 'set'. Indeed, in the most widely used version of OWL, OWL-DL, an OWL class *is* a set. 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.
Not in OWL-DL. In fact, in all the versions of OWL, a class is what it is, and has the extension that it has, in all 'situations'. There is no situation dependence in any of the semantics for any OWL dialect (and these semantics are normative, and part of the official specification.) 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.
This last statement is true for OWL-Full but not for OWL-DL. )
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).
Amen to that.
Pat Hayes
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