In several earlier posts I introduced and defended what I call Frigid Stipulation as an alternative to Rigid Designation (Introducing Frigidity, Applying Frigidity, What Kripke Really Thinks). The basic claim is that in natural languages (as opposed to in thoughts)there are no logically proper names at all, no singular terms what so ever. Every sentence with what looks grammatically like a singular term is really a disguised quantifier at the level of logical form and truth conditions and so can be analysed via Russell’s theory of descriptions.
Aside for the argument that I gave for frigidity from the fact that the truth conditions of sentences with so-called rigid designators in them change depending on who the speaker had in mind, there are also all kinds of well-known problems with construing linguistic names as logical constants, in fact with the whole idea of logical constants in the first place. These problems range from the normal ones about identity and existence statements, and belief attributions involving co-referential terms, to the bizarre logical result that we can prove that any given individual exists as a matter of first-order logic. The proof is actually quite simple and takes the form of a reductio of the assumption that the individual in question does not exist.
Here is a version of the proof that Rosenthal once presented in a Quine class I had with him.
(1) Proof by reductio that Saul Kripke exists: ((Ex) (x=SK))
1. –(Ex) (x=SK) assumption for reductio
2. (x) –(x=SK) equivalent to 1.
3. (x) (x=x) axiom of identity
4. (SK=SK) UI of 3.
5. –(SK=SK) UI of 2.
6. (SK=SK) & -(SK=SK) 4., 5.
As you can see, we derive the contradiction that Saul Kripke is both self-identical and not self-identical from the assumption that he does not exist and the axiom of identity with just two uses of universal instantiation. So we can prove that any given object exists as a matter of first-order logic with identity. But surely that is absurd! We may be able to live with the result that some object or other exists (Ex (Fx)), which naturally follows in standard first-order logic, but we cannot live with the fact that we can prove that any given particular object exists.
Even worse it seems to me that we can give an analogous proof that the object in question necessarily exists!
(2) Proof that Saul Kripke necessarily exists: □Ex (x=SK))
1. ◊ –Ex (x=SK) assumption for reductio
2. ◊ (x) –(x=SK) equivalent to 1.
3. (x)□ (x=x) modal axiom of identity
4. □ (SK=SK) UI of 3.
5. ◊ -(SK=SK) UI of 2.
6. –□ (SK=SK) equivalent to 5.
7. □ (SK=SK) & -□ (SK=SK) 4,6
But surely this is even more absurd than the last! How can I necessarily exist? These kinds of results offer good reason to adopt frigidity.