Some comments on postings by Doug Foxvog and John Sowa:
[DF] > These axioms apply to different types of set. If the FO is to
include the different set theories, then it would distinguish different
subclasses of fo:Set, e.g., fo:KPSet, fo:ZFSet, fo:KPUSet, fo:NFSet, etc.
The different axioms would apply to the appropriate subclasses of set. Then
mappings would be established between sets as defined in external ontologies
(e.g., sumo:Set) and the appropriate subclass of fo:Set. (01)
That is my intuition, but I do not know how to prove that *every* pair of
incompatible theories can be specified by axioms using some common set of
agreed terms. To make the FO project worth funding, don't think it is
necessary to prove that *mathematically*, but if we can conclude that
exceptions would be rare, I think that would provide a case for a serious
effort to try the "primitives" route to semantic interoperability. And if
not, an FO might still be stable enough for practical use - it would just
have to be tested to see how it works in practice. (02)
[JS] > >
> As soon as you add more axioms to a theory, the "meaning" of the
> so-called "primitives" changes.
>
I am not certain that that is true. If one adds subtypes to the types of
an ontology, and each subtype has some properties or restrictions not
applying to the parent, then it does not seem to me that the *meaning* of
any of the parents changes, though we are asserting more information about
the properties of the parents (i.e. that some instances have or may have
certain properties). I would not consider that a change in meaning. If we
discover a new animal that does not have any special properties other than
being a species different from other known animals, does that change the
"meaning" of the term animal? Would an inference engine be able to conclude
more inferences about the parent - or some instance of the parent not
specified as being one of the new subtypes? (03)
[JS] > > You could call subsetOf and elementOf primitives, but they don't
> behave the way that you have been claiming for the kinds of
> primitives you want. In particular, their "meaning" is determined
> by the axioms and each version of set theory has a different set
> of axioms.
>
If the logical inferences for those relations holding differ in different
theories, it would seem to me that those are different relations. I would
need specific examples to be able to see what you mean by a "different
meaning", to see how to handle such cases. (04)
Pat (05)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (06)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
> bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F. Sowa
> Sent: Wednesday, February 03, 2010 10:48 PM
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Foundation ontology, CYC, and Mapping
>
> Pat and Chris M,
>
> PC> Thanks, that is getting closer to specifics, but I am still unclear
> > exactly where the logical inconsistencies lie.
>
> The inconsistencies lie in the choice of axioms. All versions of set
> theory are based on two dyadic relations: subsetOf and elementOf.
> The differences lie in the axioms that are asserted in each theory.
>
> You could call subsetOf and elementOf primitives, but they don't
> behave the way that you have been claiming for the kinds of
> primitives you want. In particular, their "meaning" is determined
> by the axioms and each version of set theory has a different set
> of axioms.
>
> That is one of the main reasons why I keep saying that this search
> for primitives is misguided. It's totally irrelevant what set of
> words (or predicates or relations or types or concepts or whatever)
> you start with -- because all the serious work is done by the axioms.
>
> As soon as you add more axioms to a theory, the "meaning" of the
> so-called "primitives" changes.
>
> John
>
>
>
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