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## Re: [ontolog-forum] Ontology of Rough Sets

 To: "[ontolog-forum]" "John F. Sowa" Thu, 20 Jan 2011 10:20:52 -0500 <4D3852D4.9060108@xxxxxxxxxxx>
 ```The discussions in the OMG forum raised an old puzzle about two different ways of defining circles and ellipses. Euclid's way makes circles a special case of ellipses, but some programmers might define them as two independent classes.    (01) Examples like this arise in any ontology. What happens when different people define two types with specifications that might sometimes, but not always coincide?    (02) John    (03) -------- Original Message -------- Subject: Re: [architecture-strategy] Relationship between types, classes and sets Date: Thu, 20 Jan 2011 10:08:43 -0500 From: John F. Sowa To: architecture-strategy@xxxxxxxxxxxxxxx    (04) > It would help things along is you could you explain what you see the > predicates for the OO uses of Circle and Ellipse in > http://en.wikipedia.org/wiki/Circle-ellipse_problem are. > And what you see the interpretation of OO inheritance may be, where this > shows circles inheriting from ellipses - where mathematicians would expect > ellipses to be sub-types of circles.    (05) The puzzle about circles and ellipses does not violate the following criterion:    (06) Every class C has a predicate isinC(x), which is true iff object x is in class C.    (07) As for the circle-ellipse issue, it is misleading to say that one definition is more "mathematical" than the other. Both are equally mathematical (in the sense that they are defined by mathematical specifications).    (08) But mathematicians since Euclid have preferred to say that a circle is a special case of an ellipse because that assumption simplifies and generalizes the theorems and proofs. But the other definition could be assumed if anybody found it useful for some purpose.    (09) Solution: There is a trivial solution to the so-called problem that is not mentioned in the Wikipedia: drop the requirement that two isomorphic figures must belong to the same classes. That would imply that no stretched ellipse could ever *be* a circle -- the specification would take priority over the appearance in determining class membership.    (010) Re mutators: If you define a circle as having one center, and an ellipse as having two foci, you don't get any problems with mutators or inheritance. By that definition, a stretched ellipse with both foci at the same point would not be a circle, even though it looked like a circle.    (011) Re inheritance: Since no circle would ever be an ellipse, no circle could inherit two foci from the definition of ellipse.    (012) Euclid's definition is more "elegant" and "general", but elegance and generality are not prerequisites for being mathematical.    (013) John    (014) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (015) ```
 Current Thread Re: [ontolog-forum] Ontology of Rough Sets, (continued) Re: [ontolog-forum] Ontology of Rough Sets, Chris Partridge Re: [ontolog-forum] Ontology of Rough Sets, Edward Barkmeyer Re: [ontolog-forum] Ontology of Rough Sets, Chris Partridge Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, Jim Rhyne Re: [ontolog-forum] Ontology of Rough Sets, Obrst, Leo J. Re: [ontolog-forum] Ontology of Rough Sets, doug foxvog Re: [ontolog-forum] Ontology of Rough Sets, doug foxvog Re: [ontolog-forum] Ontology of Rough Sets, Christopher Menzel Re: [ontolog-forum] Ontology of Rough Sets, Pat Hayes Re: [ontolog-forum] Ontology of Rough Sets, John F. Sowa <= [ontolog-forum] Ontology of Elipses [was: of Rough Sets], doug foxvog Re: [ontolog-forum] Ontology of Rough Sets, John F. Sowa