A clarification related to Saul Kripke's argument for posteriori necessity

Question

Main argument for a posteriori necessity uses these premises:

(P1) 'Hesperus' is a proper name that refers to the evening star. 'Phosphorus' is also a proper name and it refers to the morning star. But the evening star and the morning star are the same planetary body (Venus). So both names designate Venus.

(P2) If both names designate rigidly, they designate the same object (Venus) in every possible world. Therefore (by the definition of 'necessary') 'Hesperus = Phosphorus' is necessarily true. If it is the case that in all possible worlds the identity claim “Hesperus is Phosphorus” is true, the statement is necessary.

'Hesperus' and 'Phosphorus; are proper names that we give to the evening and morning stars in all possible worlds.There are a number of problems I see with this argument; for example:

1) The object (an eveing star/morning star) might not even exist in some possible world.

2) It is possible to have an evening as well as morning star but they could be different. It is only true in our world that evening and morning stars turned out to be the same planet. It doesn't follow they would be same in all possible worlds.

What is going on here?

Answer 1

See Rigid Designators :

Kripke famously argues that because a rigid designator designates the same object in all possible worlds, an identity statement in which both designators are rigid must be necessarily true if it is true at all.

‘Hesperus = Phosphorus’ is necessarily true if true at all because ‘Hesperus’ and ‘Phosphorus’ are proper names for the same object. Like other names, Kripke maintains, they are rigid: each designates just the object it actually designates in all possible worlds in which that object exists, and it designates nothing else in any possible world. The object that ‘Hesperus’ and ‘Phosphorus’ name in all possible worlds is Venus.

a rigid designator designates its designatum in every possible world containing the designatum and in other possible worlds the designator fails to designate. In places, Kripke suggests that this is his idea:

When I use the notion of a rigid designator, I do not imply that the object referred to necessarily exists. All I mean is that in any possible world where the object in question does exist, in any situation where the object would exist, we use the designator in question to designate that object. In a situation where the object does not exist, then we should say that the designator has no referent and that the object in question so designated does not exist (Kripke, “Identity and Necessity”, 1971, p. 146).

Answer 2

• Regarding your first question, you are perfectly right: if N is a rigid designator of an entity a, which according to Kripke all proper names are, it refers to a at every world in which a exists. At the a-less worlds N doesn't refer.

• An important point that applies to proper names only and not rigid designators in genera, but which needs to be kept in mind when thinking about "Hesperus": It is our N that doesn't refer. If speakers of an a-less world use a name that is, say, orthographically indistinguishable from N, that doesn't make it N. Names are not just strings of characters or sounds, but more like institutions, individuated by name-using practices. In typical cases, one can think of these practices as inaugurated by a baptism of sorts: a few first instances of use of N, with the intention to refer to a; and then a chain of speakers which pick up the name from previous links in the chain, and form the (tacit) intention to conform to whatever name-using practice the name belongs in.

  When thinking of these questions it is important to keep straight the difference between

  1. The names that we, speakers of the actual world, use to talk about states of affairs at the counterfactual worlds, and

  2. The names that they, speakers that inhabit the counterfactual worlds, use to describe the states of affairs at their world, which they consider as actual. Their names might be homophonic to our names but they are different names, because they correspond to wholly other name-using practices.

• Regarding your second question, the whole idea of rigid designators is that they are not synonymous with any description---or any non-rigid description at least, which is the case with 'the first star visible in the morning'.

  In the case of 'Phosphorus', for example, the following sentence makes perfect sense, and is probably true:

  Phosphorus might not have been the first star visible in the morning.

  But, if 'Phosphorus' was synonymous with that description, this sentence would be a contradiction in terms. Mutatis mutandis for any other non-rigid description.