<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div>More specifically, t<span class="Apple-style-span" style="font-family: monospace; ">he standard set theoretic semantics of e.g.,</span></div><div><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; ">(\geq n R)</span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; ">i.e.,</span><span class="Apple-style-span" style="font-family: monospace; "> </span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; ">(\geq n R)^I = \{ x | #\{ y \in \Delta^I | (x,y) \in R^I \} \geq n\}</span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; "><br></span><span class="Apple-style-span" style="font-family: monospace; ">where we usually write that #S is the "cardinality of S" may be somewhat troubling (unless we use of continuum hypothesis, axioms of choice ...).</span></div><div><br></div><div><br></div><div>If we look at the FOL rewriting of concept <span class="Apple-style-span" style="font-family: monospace; ">(\geq n R), </span></div><div><span class="Apple-style-span" style="font-family: monospace; "><br></span></div><div><span class="Apple-style-span" style="font-family: monospace; ">(\geq n R)(x) = \exists_n y. R(x,y)</span></div><div><span class="Apple-style-span" style="font-family: monospace; "><br></span></div><div><span class="Apple-style-span" style="font-family: monospace; ">then I suggest the equivalent set theoretic expression </span></div><div><span class="Apple-style-span" style="font-family: monospace; "><br></span></div><div><span class="Apple-style-span" style="font-family: monospace; ">(\geq n R)^I = \{ x | \exists S \subset \{ y \in \Delta^I | (x,y) \in R^I \} such that #S = n\}<br></span></div><div><br></div><div><br></div><div>Have a nice weekend,</div><div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>-Umberto Straccia</div><div><br></div><br><div><div>On Mar 9, 2012, at 16:50 , Umberto Straccia wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Dear Colleagues,<div>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?</div><div><br></div><div>Cheers,</div><div><br></div><div>-Umberto Straccia</div><div><br></div><br><br><div>
<span class="Apple-style-span" style="border-collapse: separate; font-family: Courier; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; font-size: medium; "><span class="Apple-style-span" style="border-collapse: separate; font-family: Courier; font-size: 10px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; font-family: Monaco; font-size: 10px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; font-family: Monaco; font-size: 10px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; -webkit-text-decorations-in-effect: none; text-indent: 0px; -webkit-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div> ---------------------------------------------------</div><div> | Umberto Straccia, PhD |</div><div> | ISTI | </div><div> | Italian National Research Council |</div><div> | Via G. Moruzzi,1 |</div><div> | I-56124 Pisa (PI), ITALY | </div><div> | ------------------------------------------------ |</div><div> | WWW : <a href="http://www.umberto-straccia.name/">http://www.umberto-straccia.name</a> |</div><div> | E-mail: <a href="mailto:Umberto.Straccia@isti.cnr.it">Umberto.Straccia@isti.cnr.it</a> |</div><div> / ) Phone : +39.050.315 2894 (\</div><div> / ) Fax : +39.050.315 3464 ( \</div><div> _ ( (|___ ___________________________________________ ) )_ </div><div> (((\ \) / ) / ) / /)))</div><div> (<a href="smb:////">\\\\</a> \_/ / \ \_/ ////) </div><div> \ / \ / </div><div> \ _/ \_ / </div><div>-----/ /---------------------------------------------\ \--------</div><div> / / \ \ </div></div><br class="Apple-interchange-newline"></span></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></span><br class="Apple-interchange-newline">
</div>
<br></div></blockquote></div><br></body></html>