[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