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).
Intuitively, rigid designation is needed on the part of both designators if there are to be true "theoretical identity statements": statements in which a designator designates by way of expressing explicitly in some manner one of the designatum's theoretically interesting essences.
In other places, Kripke seems to have in mind another account of rigidity: one according to which a rigid designator designates its object in every possible world, whether or not the designatum exists in that world.
Kripke (1980) famously argues that because a rigid designator designates the same object in all possible worlds, an identity statement in which the identity sign is flanked by two rigid designators must be necessarily true if it is true at all, even if the statement is not a priori.
A rigid designator designates the same object in all possible worlds in which that object exists and never designates anything else.
Rigid designation requires that the object designated by a rigid designator be the same in all worlds.
Some authors maintain that there are so many properties that allegedly non-rigid designators for properties always end up rigidly designating some property or other in the plenum: for example, 'the element most discussed in philosophy', a candidate non-rigid designator for gold, may be said to designate rigidly the property of being the element most discussed in philosophy.
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).
'The length of S (at time t0)' is, on the other hand, a non-rigid designator for one meter; in this world it designates one meter but in other worlds, those in which S is heated or cooled, 'the length of S (at time t0)' designates other lengths.
Another theory of reference that was named about the time 'rigid designator' was coined, and that is widely associated with rigid designation, is the causal theory of reference.
However, if one designator in a true identity statement but not the other rigidly designates an entity like H2O diagonally, then the statement is a posteriori, as 'Water = H2O' is.
