Let me preface this by saying that I am willing to be convinced I am
wrong about most everything I think I understand about ontologies. I
would also like to say that I have put a lot of effort into trying to
understand, not over two days, but over nearly two years now. Maybe I'm
slow, but I managed to earn a PhD in something, just not set theory or
mathematical logic.
The part of John's definition that I am questioning is the phrase "a
class is a set that ...". I've spent two years coming to the
realization that a class is not a set, and that this idea is extremely
important for ontologies, so if you want me to return to my starting
point and throw out everything I thought I had understood, I'm going to
raise a few objections and demand evidence.
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.
Thanks for the clarification. The reference says a class is a
generalized set, and I didn't realize that meant "sets plus other
things", although now that you tell me it makes sense.
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
Thanks for this link, I think ... that's some pretty heavy going. But
what I read leads me to believe that it is open to
interpretation.
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.
I looked at every occurrence of the word "class" at this link, and I
don't see anywhere that it says that a class is a set, or not a set, or
a collection, or a .... The extension of a class consists of
individuals, fine ... so the extension is a set, but that doesn't make
the class a set. They do say "the class extension" several times,
suggesting there is only one extension, but only in cases where, if I
understand it correctly, the extension is expected to be unique, such
as The class
extension of owl:Thing,or
The class
extension of owl:Class comprises the classes of the OWL
universe. But when they talk about interpretations, they
use "a" : "a D-interpretation
of V" .
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.)
I don't see anything in the OWL semantics that would prohibit multiple
interpretations, indexed however one cares to do so, so it seems to me
that it will just happen, or not, as a matter of implementation. Let
me explain ...
In a practical implementation, where a business maintains an ontology
with their employee data in it, there are going to be modifications.
They might make a new version of the entire ontology whenever an
employee gets hired or fired, in which case you could say that you have
an entirely new ontology, with new classes, each one having their own
new, unique extension, and also new URI's that includes a
different version number. But they doesn't seem very likely to happen,
nor does it seem like a good idea.
What seems more likely is that there would be one or more TBox
ontologies, with the class definitions, and one or more ABox
ontologies, with the employee data. So someone getting hired/fired
necessitates a change to an ABox, and in a good record-keeping system,
a new version number and thus a new URI for the ABox. Even without
versions numbers, the class has a different
extension - a new set of URI's, but the class URI hasn't changed, so
it's the same class.
I realize I am making the assumption that a URI can only (ever) refer
to one thing (when properly implemented). If that assumption is flawed,
then the foundations become very shaky indeed.
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.
But the guide does not specifically say what a class is, other than
"part of the 'OWL universe'", has an extension and a member of the
extension of owl:Class. Do any of these things tell us what a class is?
And they are careful to say things like
"class
extension of all datatypes must be subsets of LVI"
so only extensions are referred to as subsets. It suggests to me that a
class is something other than a set. And a logical place to look for
what a class actually is, would be in a mathematical theory that
introduces the term, and is relevant to the issue. ZF doesn't need the
term class, so another set theory that does use the term "class" seems
reasonable. Are there other theories that use the term class in a
different way?
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.
And in model theory, don't we hear over and over again that only the
extension is a set, there are multiple interpretations Or have I been
reading the wrong references?
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.
This is what we must resort to when things are not precisely defined,
but I hope we can communicate about them so that misunderstandings, or
reasonable differences in interpretations, can get sorted out.
The prospects for genuine, shareable knowledge representation
in terms of such a language are dim at best.
I am starting to wonder ...
Chris Menzel
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