[DL] Semantics of Number restriction: small issue (?)

[Umberto Straccia]

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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