On Jan 17, 2011, at 5:00 PM, Rich Cooper wrote:
Hi Azamat,
Only the individuals and instances are
real...
You appear simply to be asserting your own personal views as if they were self-evident, eternal truths. The whole issue of what is or isn't "real" is a tremendously vexed issue and it's far from clear that any deep metaphysical commitment to this or that fundamental philosophical ontology has any useful role to play in ontological engineering at all — on which point see the recent, very interesting debate between Gary Merrill and Barry Smith/Werner Ceusters. But regardless of where you come down in that debate, nearly every useful ontology in existence takes things other than individuals as "real" — notably, the classes that populate any OWL ontology.
Perhaps you prefer that I
use the term “classes” instead of the term “sets” in
that no class can contain itself as an element.
The class THING (or whatever you call the most general class) contains itself as an element in many ontologies. And I thought you just said that only individuals and "instances" (whatever those are supposed to be) are real. What are you doing talking about classes?
That constraint leaves us free to use the
individuals, instances of data types, as the model of ground truth. In
that representation, classes are groups of individuals, instances, et
cetera. Using that representation, the equivalence function could be composed
on any or all of the four class types – rough, fuzzy, probabilistic and
crisp.
This is much too vague for anyone (well, me) to know what you are claiming. (This is not a request for more information.)
In my opinion, it is a mistake to construe
sets as containing themselves,
This is like saying it is a mistake to construe numbers as "negative". It is obviously not a "mistake" to allow non-well-founded sets; simply broaden the universe of sets beyond the well-founded sets of ZF set theory and they are (provably) just as respectable, theoretically. There are, moreover, many useful applications of non-well-founded sets. Perhaps you prefer not to avail yourself of non-well-founded sets; knock yourself out. Others find them useful.
though mathematicians have invested heavily into
that interpretation.
There are a relative few mathematicians who study them and others who have found useful applications of them. I know of no reasonable sense in which "mathematicians have invested heavily" in them.
HTH,
-chris