 It is generally acknowledged that rigidity cannot be identified with direct reference. That is because some expressions designate rigidly by means of describing the designatum: e.g., ‘The successor of 2’, which rigidly designates 3. Kripke calls designators like ‘The successor of 2’ rigid de facto, rather than rigid de jure: the description happens to be satisfied by the same object in every possible world and never anything else. Compare the intuitively distinct case of de jure rigidity in a name, like ‘Biden’. Here the intent is to refer to this person in all possible worlds, whatever descriptions may designate him.  Still, not all rigid de jure expressions seem to be directly referential (as some direct reference theorists agree: e.g., Kaplan 1989b, p. 577n.25; cf. Salmon 2003, pp. 486–7; see also Stanley’s 1997a discussion: 570–1).  An example of a description that is rigid de jure might be ‘The famous humanist most closely associated, in α, with the Italian Renaissance’, where ‘α’ is a name for the actual world (on world-indexing, see Plantinga 1974, pp. 62–65), or a similar actualized description.[6]  The designatum must first be picked out in α by whether it meets the descriptive requirements (with or without the indexing) there in α: but the purpose of the indexing is to keep the term referring, with respect to other worlds, to that same item picked out originally in α by virtue of meeting the right requirements there.  So the rigidity seems de jure.