On 8 Jul 2012, at 03:52, Pat Hayes wrote:
On Jul 7, 2012, at 12:54 PM, Chris Mungall wrote:
On Jul 6, 2012, at 7:08 PM, Pat Hayes wrote:
On Jul 6, 2012, at 4:25 PM, Chris Menzel wrote:
On Fri, Jul 6, 2012 at 4:05 PM, Michael Brunnbauer <brunni@xxxxxxxxxxxx> wrote:
On Fri, Jul 06, 2012 at 09:35:02PM +0100, Matthew West wrote:
CM> ... classes are extensional in OWL.
Is that extensional in that the extension is the members declared in the OWL ontology, or is that extensional in the sense that the members define the class, but I might not know about all of them?
I think it's extensional in the sense that classes are not first class entities
but defined via the extension of the rdf:type property.
Actually, yes, there is an RDF-compatible semantics for OWL I'd forgotten about where OWL classes are simply entities that are assigned sets of individuals as their extensions. In this semantics, distinct classes can have the same "members". But IIRC in both the W3C "direct" semantics for OWL and the "model theoretic" semantics, OWL classes are simply sets of individuals.
Pat will probably jump in here and straighten me out...
(Back from being a builder of kitchens, Pat reads lots of emails...)
FIrst, there are several OWLs. OWL-Full is the most RDF-compatible, with very few restrictions on what can be said in it, but has no complete reasoners so isn't very widely used. OWL-DL has many restrictions. OWL-Full follows RDF and RDFS in treating classes as first-class (sorry about the pun) entities and intensional, not extensional (in the sense that classes are not identified with sets, so it is consistent for two classes to have exactly the same members but still be distinct classes.) OWL-DL is quite different: it does not allow classes to be first-class entities, and it assumes that classes are defined extensionally, i.e. are sets, ie defined by their membership. So, to sum up:
extensional = classes are identified with the sets of their members.
intensional = not extensional, so having the same members does not guarantee identity of classes. (Put another way, classes have 'robust identity' which is independent of their membership.)
OWL-Full: classes are individuals, just as in RDF and RDFS and Common Logic. Classes are intensional.
OWL-DL: classes are not individuals, and properties (binary relations) only relate individuals, not classes. In the language of the ISO Common Logic specs, OWL-DL is a segregated dialect. Classes are extensional.
To be pedantic - in OWL-DL there are object properties (individual to individual), data properties (individuals to literals) and annotation properties (these are invisible in the direct semantics, but in practical terms these can link classes, provided you don't need inferences from them)
Regarding classes being the same as their extents in OWL: I don't think this view is universally shared.
Well, I havnt checked the OWL2 specs in detail, I confess, but it is certainly true in the original OWL-DL, stated quite explicitly in the semantics. Mathematical statements in a normative specification are, fortunately, not "views" to be shared or not, at will.
The OWL 1 Language Reference says:
Classes provide an abstraction mechanism for grouping resources with
similar characteristics. Like RDF classes, every OWL
class is associated with a set of individuals, called the class
extension. The individuals in the class extension are called the
instances of the class. A class has an intensional meaning
(the underlying concept) which is related but not equal to its class
extension. Thus, two classes may have the same class extension, but still be
So, if "Classes are extensional" means two OWL 1 classes with the same extent are the same class, then clearly OWL 1 classes, while having extents, are not extensional - or else this paragraph in the OWL 1 LR is wrong. FWIW I checked the errata and this paragraph is not mentioned so it seems to stand as-is.
The OWL 2 new features document claims "More importantly, backwards compatibility with OWL 1 is complete, both syntactically and semantically." even though I can't find any mention of the intensional meaning vs. class extension relationship in any of the OWL 2 documents. So what does Pat's "assumption of extensionality" mean wrt OWL 1 and OWL 2 and the question of whether two classes with the same extent are the same class?
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