Doug and Bene, (01)
The issues of types, classes, instances, and meta versions often
lead to a great deal of confusion. I'd like to mention the way
that such issues can be represented in Common Logic. (02)
BRC> I found very useful reference [3] from your first reply
> ( Instances of Instances Modeled via Higher-Order Classes
> http://www.foxvog.org/doug/higher-order2.pdf . I realized
> I tend to tangle the concept of instance/individual and
> class/meta-class but the definitions in the paper together
> with the ontology of levels of meta-classes definitely help
> to clarify these concepts. (03)
I'll relate Doug's article to Common Logic, which can be used
to support any or all of the approaches in a very clean way. (04)
In that article Doug mentions KIF, but CL has a much simpler
and more general way of handling those issues: (05)
1. The core CL semantics is untyped, but any type or class in
logics that support them can be represented by a monadic
relation in CL. (06)
2. Both the CLIF and CGIF dialects of CL support an untyped
core notation and an extended notation that allows monadic
relations to be used to restrict the range of quantifiers.
That technique can be used to support typed or sorted logics.
It can also be used to support classes in OWL and other
logics that use some notion of 'class'. (07)
3. But CL also allows quantifiers to range over relations
and functions. When monadic relations are used to represent
types or classes, CL can refer to them by quantified variables. (08)
In the article mentioned above, Doug Foxvog discusses first-order,
second-order, ..., and variable-order classes. When those classes
are represented by monadic relations in CL, all those variations
and many others can be supported, represented, and formalized
by CL axioms. (09)
In particular, the default CL semantics does not require a strict
hierarchy of types or classes (although it permits such a hierarchy).
But it avoids Russell's paradox (and others) by a very simple method:
all the so-called paradoxical statements have the truth-value false
when stated in CL. (010)
For example, the following statement would not be paradoxical
when translated to Common Logic: (011)
T is the type of all types that are not types of themselves. (012)
In CL, that sentence would become: (013)
T is a monadic relation that is true of all monadic relations
that are not true of themselves. (014)
In CL semantics, it is false that there exists a T that satisfies
the condition. End of story. No paradox. (015)
For further discussion of these and other issues, see the
following article: (016)
http://www.jfsowa.com/cg/cg_hbook.pdf (017)
This is an article about conceptual graphs, but every example is
stated in both CGIF and CLIF. At the end, the references cite
articles by Pat Hayes and Chris Menzel about the semantics,
the IKL extensions, and the mapping of RDF and OWL to CL. (018)
John Sowa (019)
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