A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else.
First, a rigid designator designates the same object in all possible worlds as it is used in the actual world, not as it is used in other possible worlds in which the object gets picked out: for although we identify objects in other worlds by our own names, natives of some of these worlds use other names (Kripke 1980, p. 77).
However, as Kripke has defined 'strongly rigid', 'Hesperus' cannot be a "strongly rigid designator"; that distinction is reserved for designators that designate a necessarily existing object (1980, pp. 48-9).
In that case, of course, broad causal grounding is ubiquitous and not specially tied to rigidity: all singular concrete object designators, including non-rigid definite descriptions, are broadly causally grounded.
Under the stipulations that admissible models make all individual constants rigid designators and that I P,w) ⊆ Don, the standard free logic PFL, together with the modal axioms and rules appropriate to whatever structure we assign to R, is sound and complete on this semantics.
"All we got was standard designators and lot numbers and such".
Kripke would say that if both designators in an identity statement rigidly designate the same entity, then the statement is necessarily true, as 'Water = H2O' is.
In order to deal with negative propositions, Armstrong Armstrongg 1997) invokes a certain kind of facts which he calls 'totality facts', facts which can be designated by means of expressions of type 'the fact that X are all the Fs that there are', where 'X' is a rigid plural designator and 'F' a predicate.
Such a semantics also requires free logic when Do contains objects not in U, for in that case rigid designators of these objects are empty in all worlds.
But he argued that when the terms flanking the sign of identity were what he called rigid designators, an identity statement, if true at all, had to be necessarily true, but need not be knowable a priori, as an analytic truth would be.
