Peter Geach and Elizabeth Anscombe were younger colleagues of Ludwig Wittgenstein. Geach tried to synthesize analytic philosophy and Thomism. He was an expert on Gottlob Frege. With Max Black, in 1952 he translated Frege's works.

For Geach and Wiggins, relative identity means "x is the same F as y," but "x may not be the same G as y." Wiggins argued against this idea of relative identity, but accepted what he called a sortal-dependent identity, "x is the same F as y." Geach called this a "criterion of identity."

I had here best interject a note on how I mean this term "criterion of identity". I maintain that it makes no sense to judge whether x and y are 'the same', or whether x remains 'the same', unless we add or understand some general term—"the same F". That in accordance with which we thus judge as to the identity, I call a criterion of identity; this agrees with the etymology of "criterion". Frege sees clearly that "one" cannot significantly stand as a predicate of objects unless it is (at least understood as) attached to a general term; I am surprised he did not see that the like holds for the closely allied expression "the same".

(Reference and Generality, 1962, p.39; 1980, p.63)

I am arguing for the thesis that identity is relative. When one says "x is identical with y", this, I hold, is an incomplete expression; it is short for "x is the same A as y", where "A" represents some count noun understood from the context of utterance—or else, it is just a vague expression of a half-formed thought. Frege emphasized that "x is one" is an incomplete way of saying "x is one A, a single A", or else has no clear sense; since the connection of the concepts one and identity comes out just as much in the German "ein und dasselbe" as in the English "one and the same", it has always surprised me that Frege did not similarly maintain the parallel doctrine of relativized identity, which I have just briefly stated. On the contrary, Frege actually enunciated with all vigour a doctrine that identity cannot be relativized: "Identity is a relation given to us in such a specific form that it is inconceivable that various forms of it should occur" (Grundgesetze, Vol. II, p. 254).

Absolute identity seems at first sight to be presupposed in the branch of formal logic called identity theory. Classical identity theory may be obtained by adjoining a single schema to ordinary quantification theory (for bound name-variables):

⊢Fa ↔ Vx(Fx ∧ x=a).         (1)

Quine in his Set Theory and its Logic attributes to Hao Wang (p. 13) the recognition that (1) will serve as a single axiom schema for identity theory. In the vernacular we may intuitively express the content of (1) by saying: Whatever is true of something identical with an object a is true of a, and conversely. We readily derive from schema (1) the Law of Self-Identity, "⊢(a = a)". For if we take "Fξ" to be "ξ ≠ a", then schema (1) gives us:

⊢(a ≠ a) ↔ Vx(x ≠ a) ∧ x = a),        (2)

It was Ruth Barcan Marcus who introduced the identity of indiscernibles

which of course yields "⊢a = a". And there are equally easy proofs of what has been called (I believe by Quine) the Indiscernibility of Identicals:

⊢Fb ∧ b = a → Fa           (3)

and of theorems asserting the symmetry and transitiveness of identity as a relation. The logical system got by adjoining schema (1) to classical quantification theory is a system with a complete proof procedure; moreover, its interpretation is categorical, in the following sense: If we try to introduce two two-place predicables, each separately conforming to schema (1), they turn out to coincide in application. (In this paper, as in my book Reference and Generality, I use "predicables" as a term for the verbal expressions called "predicates" by other logicians; I reserve the term "predicate" for a predicable actually being used as the main functor in a given proposition.)

(Logic Matters, 1972, pp.238-239)

In 1968, David Wiggins described Geach's first version of Tibbles. Where Theon is identical to Dion except he is missing a foot, we now have a cat named Tibbles and a second cat named Tib who lacks a tail.

Wiggins begins his argument with an assertion S*

S*: No two things of the same kind (that is, no two things which satisfy the same sortal or substance concept) can occupy exactly the same volume at exactly the same time.

This, I think, is a sort of necessary truth...

A final test for the soundness of S* or, if you wish, for Leibniz' Law, is provided by a puzzle contrived by Geach out of a discussion in William of Sherwood. A cat called Tibbles loses his tail at time t₂. But before t₂somebody had picked out, identified, and distinguished from Tibbles a different and rather peculiar animate entity-namely, Tibbles minus Tibbles' tail. Let us suppose that he decided to call this entity "Tib." Suppose Tibbles was on the mat at time t₁. Then both Tib and Tibbles were on the mat at t₁. This does not violate S*.

But consider the position from t₃ onward when, something the worse for wear, the cat is sitting on the mat without a tail. Is there one cat or are there two cats there? Tib is certainly sitting there. In a way nothing happened to him at all. But so is Tibbles. For Tibbles lost his tail, survived this experience, and then at t₃ was sitting on the mat. And we agreed that Tib ≠ Tibbles. We can uphold the transitivity of identity, it seems, only if we stick by that decision at t₃ and allow that at t₃ there are two cats on the mat in exactly the same place at exactly the same time. But my adherence to S* obliges me to reject this. So I am obliged to find something independently wrong with the way in which the puzzle was set up.

This is a clear case of Peter van Inwagen's Doctrine of Arbitrary Undetached Parts

It was set up in such a way that before t₂ Tibbles had a tail as a part and Tib allegedly did not have a tail as a part. If one dislikes this feature (as I do), then one has to ask, "Can one identify and name a part of a cat, insist one is naming just that, and insist that what one is naming is a cat"? This is my argument against the supposition that one can: Does Tib have a tail or not? I mean the question in the ordinary sense of "have," not in any peculiar sense "have as a part." For in a way it is precisely the propriety of some other concept of having as a part which is in question.

As an arbitrary undetached part, Tib has been picked out and defined as coinciding with Tibbles, except for the tail Tibbles is about to lose. This violates S*

Surely Tib adjoins and is connected to a tail in the standard way in which cats who have tails are connected with their tails. There is no peculiarity in this case. Otherwise Tibbles himself might not have a tail. Surely any animal which has a tail loses a member or part of itself if its tail is cut off. But then there was no such cat as the cat who at t₁ has no tail as a part of himself. Certainly there was a cat-part which anybody could call "Tib" if they wished. But one cannot define into existence a cat called Tib who had no tail as part of himself at t, if there was no such cat at t₁. If someone thought he could, then one might ask him (before the cutting at t₂), "Is this Tib of yours the same cat as Tibbles or is he a different cat?"

("Being in the same place at the same time,"The Philosophical Review, p.94)

In Geach's second account of Tibbles as an exemplar of a metaphysical problem, published some years later (1980), Tibbles is a cat with 1,000 hairs that can be interpreted as 1,001 cats, by "picking out" and then pulling out one of those cat hairs at a time and each time identifying a new cat..

Geach's second version of Tibbles is widely cited as a discussion of the problem of vagueness or what Peter Unger called the Problem of the Many, also published in 1980. It is not the "body-minus" problem of the original Tibbles.

If a few of Tibbles' hairs are pulled out, do we still have Tibbles, the Cat? Obviously we do. Have we created other cats, now multiple things in the same place at the same time? Obviously not.

Geach argues that removing one of a thousand hairs from Tibbles shows that there are actually 1,001 cats on the mat.

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 h_1,000.

Now let c be the largest continuous mass of feline tissue on the mat. Then for any of our 1,000 cat-hairs, say h_n, there is a proper part c_n of c which contains precisely all of c except the hair h_n; and every such part cn differs in a describable way both from any other such part, say c_m, 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 c_n is a cat: c_n would clearly be a cat were the hair h_n plucked out, and we cannot reasonably suppose that plucking out a hair generates a cat, so c_n 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!

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, . . . c_1.000 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.

(Reference and Generality, 3rd edition, p.215.)

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