Thank you, Bart.
It was a very good try, but its solution is hardly within the orbit of
formalization. We have here two different sorts of things: logical entities
( a "species" relates to an idea, and an "individual", to a reality, so an
instanceofrelation is of different status than a subclassofrelation.
Possiblly, here "the dog burried." (01)
There is a widelyused hierarchy down to an individual, like in Richard's
notations:
begin hierarchy example;
animal;
// dog;
/// Fido;
end hierarchy example;
Offline I asked Bart how it is resolved in the automated reasoning of
ontologies, and if it is mixed with the class inclusion hierarchy. For there
are at least two
types of hierarchy, (add here parthood):
1. restrained by the class inclusion relationships;
2. restrained by the the class membership relationship.
As above, Fido, as an individual dog, is a member of the class of dogs, the
last one is a member of the class of species of animals. But Fido is not a
member of the latter, because an individual is never a species (of animal).
And the number of individual dogs doesn't affect the number of the species. (02)
Caveat: the class membership is not transitive, while the class inclusion is
transitive. Or, if x is an instance of class X, which is a subclass of Y,
then it is not necessarily that x is an instance of Y.
To my undertanding, most SW languages's formal semantics as a basic
assumption of reasoning have the opposite: if x is an instance of class X,
which is a subclass of Y, then it is necessarily that x is an instance of Y.
Wonder, Alan, if its the same with a new version, OWL 2 or is to be with OWL
3...?
Azamat Abdoullaev
http://www.eis.com.cy (03)
 Original Message 
From: "Bart Gajderowicz" <bgajdero@xxxxxxxxxx>
To: "[ontologforum]" <ontologforum@xxxxxxxxxxxxxxxx>
Sent: Saturday, April 25, 2009 3:54 AM
Subject: Re: [ontologforum] cyclic and acyclic definitions (04)
Thanks for the suggestion, John. My background is in computer science,
so I'm not as well versed in mathematics and logic, as the members of
the forum. I'm always learning, and the feedback has been very
helpful. (05)
Thanks again.

Bart Gajderowicz
MSc Candidate, '10
Dept. of Computer Science
Ryerson University
http://www.scs.ryerson.ca/~bgajdero (06)
2009/4/24 John F. Sowa <sowa@xxxxxxxxxxx>:
> Bart,
>
> Definitions of the following kind are trivial substitutions:
>
>> Momo = Man who has Only Male Offspring
>> (forall x (Momo(x) iff exists y s.t. Man(x) ^ Momo(y) ^ hasChild(y,x)))
>
> Following is the more common form of definition in mathematics:
>
> A dyadic relation R is transitive iff the following axiom is true:
>
> For all x, y, z,
> R(x,y) and R(y,z) implies R(x,z).
>
> More generally, the basic form of a definition in mathematics
> names the type of entity (e.g. 'transitive relation') and
> states a symbol such as R that represents an arbitrary instance.
> Then that statement is followed by one or more axioms that
> specify the constraints on any such entity R.
>
> Mathematicians have been using such definitions since Euclid,
> and they have never caused any trouble at all.
>
> Perhaps you're thinking of the following paradoxical statement
> in set theory:
>
> The set of all sets that are not elements of themselves.
>
> But that is a result of a peculiarity caused by the 'element of'
> relation. There is no paradox with the following statement:
>
> The set of all sets that are not subsets of themselves.
>
> Since every set is a subset of itself, the set defined by
> that sentence happens to be empty. No paradox.
>
> Suggestion: I recommend that we following the well established
> practice of mathematicians for stating definitions.
>
> John Sowa
>
>
>
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