On Jan 21, 2011, at 10:22 PM, doug foxvog wrote:
> On Fri, January 21, 2011 12:18, Christopher Menzel said:
>> 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.
>>
>> I think it's useful to distinguish two claims when it comes to the identity
>conditions of classes:
>>
>> (1) Classes are not extensional (i.e., distinct classes can have the same
>members/instances)
>>
>> (2) Classes can change their membership.
>>
>> In the formal semantics of a number of KR languages, (1) is true but,
>strictly speaking at least, (2) is not.
>
> Ontology ALWAYS comes up against the problem that the same word is used with
>different meanings. Meriam-Webster's 11th edition has 24 definitions for "set"
>as a noun and 6 for "class" as a noun. (01)
Sure thing. Are we disagreeing about something? :-) (02)
> I just said that this was A standard distinction, not that the word "class"
> was not used in other ways. (03)
I entirely agree; I probably wasn't clear enough, but my intention was only to
expand on your point. (04)
>> Notably, classes in OWL areexplicitly non-extensional: since a class is
>stipulated only to *have* anextension in OWL's formal semantics, nothing
>prevents distinct classesfrom having the same extension. The same is true of
>RDF.
>
> Agreed.
>
>> 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
>classchange its membership.
>
> The restriction here, "within a given interpretation", places the
> restriction of unchangability on the class. (05)
Right, that's all I was trying to point out. (06)
> A class can change its membership from one interpretation to another merely
>with the addition or removal of statements. (07)
Sure. (08)
>> As noted in an earlier message in this thread,without augmenting the notion
>of an OWL interpretation somehow, change canonly be represented formally in
>terms of something like a series ofinterpretations that are thought of as
>temporally ordered.
>
> Fine, for change that is thought of as temporal change. For change that is
>spatial/jurisdictional, the different interpretations need not be temporally
>ordered. (09)
Absolutely true, I was focusing on change through time, although even if the
focus is on spatial or jurisdictional change, it seems to me that it will
nearly always be important also to know how the change in question is
temporally ordered; otherwise all we know is that there was change, that
something was in one location/state A at one time and another location/state B
at some other time, not that something changed *from* (say) location/state A
*to* location/state B. (010)
> Such change allows for membership in a class to change, so I don't think that
>is an argument that membership in a class can not change. (011)
Just to be clear, I never argued that membership in a class cannot change. I
simply said that change of membership by an OWL class is not possible within a
given OWL interpretation. That's it. (012)
>> That said, (2) does seem to be a strong *intuitive* idea in the KR, AI, and
>database communities.
>
> This is the meaning i was referring to. (013)
Never meant to suggest otherwise! :-) (014)
> The next comment deals with the meaning of "class" in a set theoretical
> context. This is a different meaning of "class" than the ontological
> context uses. (015)
I took that to be implied by my mentioning (V)NBG after the preceding claim. (016)
>> Finally, the idea that sets are extensional and classes are not isdefinitely
>not standard among logicians and mathematicians, who typicallyassociate the
>notion of class with theories like VNBG, wherein bothclasses and sets are
>extensional.
>
> My understanding is that NBG's principle of class comprehension is
>predicative. (017)
Only with respect to *proper* classes, since the quantifiers in the formulas
that can be used in comprehension are restricted to sets. It is impredicative
if the formula used in a given instance contains quantifiers and happens to
pick out a set. (018)
> Since it is a set theory, the defining predicates of the class are static,
>and thus an NBG class would have static membership. (019)
I'm not quite sure what you mean by a predicate being static, but I would put
what is probably the same point by saying that it is the universe of sets and
classes described by NBG that is static; the notion of change simply has no
purchase therein. (020)
Regards, (021)
-chris (022)
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