 Modal semantics thus defined call for free logic whenever worlds are allowed to have differing domains—that is whenever we may have worlds \(u\) and \(w\) such that \(\bD_u \ne \bD_w\). For in that case there must be an object \(d\) that exists in one of these domains (let it be \(\bD_w)\), but not the other, so that any singular term \(t\) that rigidly designates \(d\) must be empty at world \(u\). Hence \({\sim}\exists x(x=t)\) (which is self-contradictory in classical logic) must be true at world \(u\). Such a semantics also requires free logic when \(\bD_o\) contains objects not in \(\bU\), for in that case rigid designators of these objects are empty in all worlds. Finally, this semantics calls for inclusive logic if any world has an empty domain. Thus, given this semantics, the only way to make the resulting logic unfree is to require that domains be fixed—i.e., that all worlds have the same domain \(\bD\), that \(\bD\) be non-empty, and that \(\bD_o = \bD\).