Rigidity and necessary application

The question whether the notion of rigidity can be extended in a fruitful way beyond singular terms has received a standard answer in the literature, according to which non‐singular terms designate kinds, properties or other abstract singular objects, and generalized rigidity is the same thing as singular term rigidity, but for terms designating such objects. I offer some new criticisms of this view and go on to defend an alternative view, on which non‐singular terms designate extensions in general, and generalized rigidity is identity of extension across possible worlds. I develop some fundamental positive considerations that make this view virtually inevitable as a view of generalized rigidity, emphasizing its exclusive ability to offer a purely logical justification of the necessity of several kinds of statements that go beyond true identity statements between rigid singular terms.