Kaplan's own framework distinguishes necessity (truth at all worlds) from what is merely 'dthat'-fixed: a rigidified description is not thereby metaphysically necessary (Kaplan, 'Demonstratives', 1989).
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dthat(a technical term Kaplan used in his framework)
A special operator Kaplan invented in his work on demonstratives that 'freezes' what a description refers to, locking it to a particular thing.
rigidified description(Philosophy of language; debate over whether proper names are equivalent to such descriptions)
A definite description modified (typically by the 'actual' operator) so that it designates the same individual in all possible worlds, intended to mimic the referential behavior of proper names
truth at all worlds(another way of saying something is necessary)
A statement that would be true no matter what the circumstances were—even if reality had turned out completely differently.