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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 21 Jan 2011 11:18:16 -0600
Message-id: <A5383CB4-D880-4BA1-A8A7-AF0B4A11495D@xxxxxxxx>
On Jan 21, 2011, at 9:46 AM, doug foxvog wrote:
> ...
> A standard distinction between a set and a class, is that membership in
> a [set] cannot change, while membership in a class can.      (01)

I think it's useful to distinguish two claims when it comes to the identity 
conditions of classes:    (02)

(1) Classes are not extensional (i.e., distinct classes can have the same 
members/instances)    (03)

(2) Classes can change their membership.    (04)

In the formal semantics of a number of KR languages, (1) is true but, strictly 
speaking at least, (2) is not.  Notably, classes in OWL are explicitly 
non-extensional: since a class is stipulated only to *have* an extension in 
OWL's formal semantics, nothing prevents distinct classes from having the same 
extension.  The same is true of RDF.  However, simply because there is no 
formal notion of change built into OWL's semantics, there is no possibility, 
within a given interpretation, that a class change its membership.  As noted in 
an earlier message in this thread, without augmenting the notion of an OWL 
interpretation somehow, change can only be represented formally in terms of 
something like a series of interpretations that are thought of as temporally 
ordered.  That said, (2) does seem to be a strong *intuitive* idea in the KR, 
AI, and database communities.    (05)

Finally, the idea that sets are extensional and classes are not is definitely 
not standard among logicians and mathematicians, who typically associate the 
notion of class with theories like VNBG, wherein both classes and sets are 
extensional.    (06)

-chris    (07)

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