# No subject

Sat Nov 23 22:07:52 CET 2013

of the interpretation function non-disjoint is not a problem
(it does not make the definition of the semantics non-wellfounded,
cyclic, or anything like that), and it is never explicitly
required. By the way, I just wrote a paper where in some of the
proofs I view a TBox as an interpretation of itself, and thus I
I write things like $A \in A^I$ where $A$ is a concept name.

On the other hand, this means that just making the domain and range
of the interpretation function non-disjoint does not automatically
yield meta-classes or such. Say we have concept names A, B, C
and A, B are also elements of the interpretation domain (the
range of the interpretation function). Then we may interpret
A by the set {A,B}. No cyclicity arises since we don't apply
the interpretation function again to the A or B in this set.
However, this also means that the A in the set is viewed as
an individual (that just happens to have the same name as
the concept A). This does not make A a meta-class.

Thus, if you want meta-classes, you may need to make the domain
and range of the interpretation function non-disjoint, but this
is not enough. You must also modify the definition of what an
interpretation function does appropriately. And when you do this,
then you must be careful not to produce a cyclic, non-wellfounded
definition.

Best regards,

****************** baader at inf.tu-dresden.de *******************