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

Stasinos Konstantopoulos konstant at iit.demokritos.gr
Tue Mar 13 00:07:52 CET 2012

Umberto, hi.

this seems to imply that you need a domain that is not enumerable,
otherwise you would not worry about whether

\{ y \in \Delta^I | (x,y) \in R^I \}

is enumerable or not.

But needing a non-enumerable domain sounds strange, given that enumerable
domains are sufficient for the interpretation of FOL (Löwenheim-Skolem).

Best,
Stasinos

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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