110. 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 fat 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 these 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. We simply do not speak of cats, or use names of cats, in this way; nor is our ordinary practice open to logical censure. I am indeed far from thinking that ordinary practice never is open to logical censure; but I do not believe our ordinary use of proper names and count nouns is so radically at fault as this conclusion would imply. 

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. 

Thus each one of the names "c1, c2, . . . c1,000", or again the name "c", is a name of a cat; but none of these 1,001 names is a name for a cat, as "Tibbles" is. By virtue of its sense "Tibbles" is a name, not for one and the same thing (in fact, to say that would really be to say nothing at all), but for one and the same cat. This name for a cat has reference, and it names the one and only cat on the mat; but just on that account "Tibbles" names, as a shared name, both c itself and any of the smaller masses of feline tissue like c12 and c279; for all of these are one and the same cat, though not one and the same mass of feline tissue. "Tibbles" is not a name for a mass of feline tissue.

So we recover the truth of the simple story we began with. The price to pay is that we must regard "— is the same cat as —" as expressing only a certain equivalence relation, not an absolute identity restricted to cats; but this price, I have elsewhere argued, must be paid anyhow, for there is no such absolute identity as logicians have assumed.

Peter Geach, Reference and Generality.