[DL] Semantics of Number restriction: small issue (?)
Pascal Hitzler
pascal.hitzler at wright.edu
Mon Mar 12 17:32:41 CET 2012
It would seem that this really only makes a difference if we
1) are interested in infinite n's in number restrictions (and I wouldn't
think that we need this?) and
2) want to avoid using the well-ordering principle (which is equivalent
to the axiom of choice).
Am I missing a point here?
Best Regards,
Pascal.
On 3/9/2012 12:39 PM, Umberto Straccia wrote:
> 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
>>
>>
>>
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--
Prof. Dr. Pascal Hitzler
Dept. of Computer Science, Wright State University, Dayton, OH
pascal at pascal-hitzler.de http://www.knoesis.org/pascal/
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