A two-dimensionalist adds that if both designators rigidly designate the same entity diagonally, then the statement is a priori, as 'Watery stuff = wet, drinkable … stuff' is, or as 'H2O = H2O' is.
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.
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.
A two-dimensionalist naturally understands this claim as one that concerns horizontal rigidity: for her, if both designators in an identity statement rigidly designate the same entity horizontally, then the statement is necessarily true, as 'Water = H2O' is.
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.
Some philosophers have proposed that 'Apis mellifera' rigidly designates an abstract object, a kind or property.
Uttered in a fixed context, 'I' rigidly designates, even though it is an indexical.
On the second interpretation, a name rigidly designates a given object at every world regardless of whether the object exists in that world.
For rigidly designating terms, this raises the question of how to evaluate atomic formulas at worlds in which their referents do not exist.
