How is "---- is the same A as ----" related to a proper name for an A? To attack this problem, I shall first set forth a paradox that I developed from a sophisma of William of Sherwood.
The fact cat sat on the mat. There was just one cat on the mat. The cat's name was "Tibbles": "Tibbles" is moreover a name for a cat.--This simple story leads us into difficulties if we assume that Tibbles is a normal cat. For a normal cat has at least 1,000 hairs. Like many empirical concepts, the concept (single) hair is fuzzy at the edges ; but it is reasonable to assume that we can identify in Tibbles at least 1,000 of his parts each of which definitely is a single hair. I shall refer to thee hairs as h1, h2, h3...up to h1,000.
Now let c be the largest continuous mass of feline tissue on the mat. Then for any of our 1,000 cat-hairs, say hn, there is a proper part cn of c which contains precisely all of c except the hair hn; and every such part cn differs in a describable way both from any other such part, say, cm, and from c as a whole. Moreover, fuzzy as the concept cat may be, it is clear that not only is c a cat, but also any part cn is a cat: cn would clearly be a cat were the hair hn plucked out, and we cannot reasonably suppose that plucking out a hair generates a cat, so cn must already have been a cat. So, contrary to our story, there was not just one cat called "Tibbles" sitting on the mat; there were at least 1,001 sitting there! Of course this would involve a great deal of overlap and sharing of organs among these 1,001 cats, but logic has nothing to say against that; after all, it happens on a small scale between Siamese twins.
All the same, this result is absurd....
Everything falls into place if we realize that the number of cats on the mat is the number of different cats on the mat; and c13, c279, and c are not three different cats, they are one and the same cat. Though none of these 1,001 lumps of feline tissue is the same lump of feline tissue as another, each is the same cat as any other: each of them, then, is a cat, but there is only one cat on the mat, and our original story stands.
(Peter Geach, Reference and Generality, Cornell UP [Ithaca: 1980] 215-216)
Geach then points out that this appears to commit us to the claim that "---- is the same cat as ----" is an equivalence relation, not an absolute identity relation; which, of course, he has no problem with, since he doesn't think there is any such thing as absolute identity. He also argues that it shows "that logic can and must avoid assuming a syntactical category of proper names." A proper name for an A may be the shared name of several Bs, if Bs are in some sense the same A as any of the others. An interesting argument, although I'm not sure this is a wise argument for this conclusion. I'll have to look more closely at what Geach says about proper names.
In any case, I wonder which of William's sophismata Geach has in mind, and whether he mentions it less vaguely elsewhere.