 It may be that no substantive issues ride on which conception of rigidity is adopted (Stanley 1997a, pp. 557, 566ff.). On the other hand, some philosophers have held that true statements using a proper name to express that so and so might not have existed are unintelligible unless the relevant name refers to the object in all worlds, period. That might provide a substantive reason for favoring the latter type of rigidity (see, e.g., Besson 2009; Gómez-Torrente 2006, p. 250; Plantinga 1985, p. 84; on Plantinga, see also the subsection on “Worlds and Essences” in the entry on  actualism), obstinate rigidity, as Salmon calls it (1981, p. 34).  Kripke’s quote above hints that obstinate rigidity might afford this kind of advantage. But Kripke never argues for one position or another. Despite occasional slips in favor of one or another refined version of rigidity, he deliberately sidesteps these “delicate issues” when he gives them his full attention (1980, p. 21, n. 21).