[DL] Semantics of Number restriction: small issue (?)
Umberto Straccia
umberto.straccia at isti.cnr.it
Fri Mar 9 18:39:01 CET 2012
More specifically, the standard set theoretic semantics of e.g.,
(\geq n R)
i.e.,
(\geq n R)^I = \{ x | #\{ y \in \Delta^I | (x,y) \in R^I \} \geq n\}
where we usually write that #S is the "cardinality of S" may be somewhat troubling (unless we use of continuum hypothesis, axioms of choice ...).
If we look at the FOL rewriting of concept (\geq n R),
(\geq n R)(x) = \exists_n y. R(x,y)
then I suggest the equivalent set theoretic expression
(\geq n R)^I = \{ x | \exists S \subset \{ y \in \Delta^I | (x,y) \in R^I \} such that #S = n\}
Have a nice weekend,
-Umberto Straccia
On Mar 9, 2012, at 16:50 , Umberto Straccia wrote:
> Dear Colleagues,
> it appears to me that the semantics of number restrictions concepts in DLs may need a minor fix, as the notion of "the cardinal of a set" is defined for sets that are equipollent to ordinal numbers only. Isn't it?
>
> Cheers,
>
> -Umberto Straccia
>
>
>
> ---------------------------------------------------
> | Umberto Straccia, PhD |
> | ISTI |
> | Italian National Research Council |
> | Via G. Moruzzi,1 |
> | I-56124 Pisa (PI), ITALY |
> | ------------------------------------------------ |
> | WWW : http://www.umberto-straccia.name |
> | E-mail: Umberto.Straccia at isti.cnr.it |
> / ) Phone : +39.050.315 2894 (\
> / ) Fax : +39.050.315 3464 ( \
> _ ( (|___ ___________________________________________ ) )_
> (((\ \) / ) / ) / /)))
> (\\\\ \_/ / \ \_/ ////)
> \ / \ /
> \ _/ \_ /
> -----/ /---------------------------------------------\ \--------
> / / \ \
>
>
>
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <https://mailman.informatik.uni-bremen.de/pipermail/dl/attachments/20120309/2bb39521/attachment.html>
More information about the dl
mailing list