In an earlier post I looked approvingly at Tim Crane's views on singularity, and promised to follow up with some things I don't like so much. Now what I don't like so much is Crane's characterisation of failed reference. He says that "a thinker can think about a particular object and yet fail to refer to that object in thought", and that “There are many cases where thinkers appear to be having singular thoughts in this sense even though the object of the thought does not exist: aiming to refer to a specific object in this case fails to ‘hit’ the target object" (my emphasis).
On the contrary. As I have argued here, a proper name individuates: it tells us which individual a proposition is about. It is easy to show (a) that it cannot fail to do this, once understood, and (b) that it has no other function.
(a) A proper name individuates by telling us which sentences are verified by a single subject. The sentences "Frodo is a hobbit ... Frodo has large feet" together say that some hobbit has large feet, i.e. a single thing is both a hobbit and has large feet, and not that some thing is a hobbit and that some thing (possibly a different thing) has large feet. You simply haven't understood how the proper name works if you think that both sentences could be true without being true of a single thing. I say a bit more about this here.
(b) This is all that proper names do. (i) They have no descriptive sense. They tell us which individual a sentence is about by telling us which individual it is the same as. The 'Frodo' of the second sentence above tells us that if the second sentence is true, it is true of the same thing as the first sentence, if true, is true of, and no more. This is exactly how proper names individuate in stories, and it is clear they can do no more than this. Nor can they do any more even if the story is true, and all the names 'refer'. (ii) They have no 'extra-linguistic' sense. As I argued here, if any piece of language has an important communication function, we should be able to tell whether it has an important communication function. One of the most important features of communicating with someone is that they should know they are being communicated to. Therefore, if non-empty proper names communicate information that empty proper names don’t, we should know this, and we should be able to tell whether a name was empty or not. But we can't do this. We do not know for certain whether the Christ Myth theory is true or not, and thus we don't know whether the name 'Jesus' is empty or not.
There is nothing that a proper name could try to do, that it does not do. A proper name tells us which character is being written about, and it does this successfully whether in a story ('King Arthur found a sword') or in a piece of history ('King Albert burned the cakes'). Thus Crane's idea that it could possibly fail to accomplish anything is a mistake.