The Grice-Myro Theory of Identity (Grice Archive, Carton 7, Folder 10)

From the Stanford online (Deutsche), adapted.

"The fundamental claim of TIME-relative identity - the claim the various versions of the idea have in common - is that, as it seems in the passenger/person case, it can and does happen that x and y are the same F and (yet) x and y are not the same G."

"Now it is usually supposed that if x and y are the same F (G etc.), then that implies that x and y are Fs (Gs, etc.) If so, then the above schema is trivially satisfied by the case in which x and y are the same person but x (y) is not a passenger at all. But let us resolve to use the phrase ‘x and y are different Gs’ to mean ‘x and y are Gs and x and y are not the same G’. Then the nontrivial core claim about relative identity is that the following may well be true."

RI: x and y are the same F but x and y are different Gs.

"RI is a very interesting thesis. It seems to yield dramatically simple solutions to (at least some of) the puzzles about identity. We appear to be in a position to assert that young Oscar and old Oscar are the same dog but nonetheless distinct "temporary" objects; that Oscar and Oscar-minus are the same dog but different dog parts; that the same piece of clay can be now (identical to) one statue and now another; that London and Londres are the same city but different "objects of thought," and so forth. Doubts develop quickly, however. Either the same dog relation satisfies LL or it does not. If it does not, it is unclear why it should be taken to be a relation of identity. But if it satisfies LL, then it follows, given that Oscar and Oscar-minus are different dog parts, that Oscar-minus is not the same dog part as Oscar-minus. Furthermore, assuming that the same dog part relation is reflexive, it follows from the assumption that Oscar-minus and Oscar-minus are the same dog (and that LL is in force), that Oscar and Oscar-minus are indeed the same dog part, which in fact they are not."

"It may seem, then, that RI is simply incoherent. These arguments, however, are a bit too quick. On analysis, they show only that the following three conditions form an inconsistent triad: (1) RI is true (for some fixed predicates F and G). (2) Identity relations are equivalence relations. (3) The relation x and y are the same F figuring in (1) satisfies LL. For suppose that the relation x and y are the same G, figuring in (1), is reflexive and that x is a G. Then x is the same G as x. But according to (1), x and y are not the same Gs; hence, according to (3), it is not the case that x and y are the same F; yet (1) asserts otherwise. Now, most relative identity theorists maintain that while identity relations are equivalence relations, they do not in general satisfy LL. However, according to at least one analysis of the passenger/person case (and others), the same person relation satisfies LL but the same passenger relation is not straightforwardly an equivalence relation (Gupta 1980). It should be clear though that this view is incompatible with the principle of the identity of indiscernibles: If x and y are different passengers, there must be, by the latter principle, some property x possesses that y does not. Hence if the same person relation satisfies LL, it follows that x and y are not the same person. For the remainder we will assume that identity relations are equivalence relations. Given this assumption, (and assuming that the underlying propositional logic is classical — cf. Parsons 2000) RI and LL are incompatible in the sense that within the framework of a single fixed language for which LL is defined, RI and LL are incompatible."

"Yet the advocate of relative identity cannot simply reject any form of LL. There are true and indispensable instances of LL: If x and y are the same dog, then, surely, if x is a Dalmatian, so is y. The problem is that of formulating and motivating restricted forms of LL that are nonetheless strong enough to bear the burden of identity claims. There has been little systematic work done in this direction, crucial though it is to the relative identity project. (See Deutsch 1997 for discussion of this issue.) There are, however, equivalence relations that do satisfy restricted forms of LL. These are sometimes called ‘congruence relations’ and they turn up frequently in mathematics. For example, say that integers n and m are congruent if their difference n − m is a multiple of 3. This relation preserves multiplication and addition, but not every property. The numbers 2 and 11 are thus congruent but 2 is even and 11 is not. There are also non-mathematical congruencies. For example, the relation x and y are traveling at the same speed preserves certain properties and not others. If objects x and y are traveling at the same speed and x is traveling faster than z, the same is true of y. Such similarity relations satisfy restricted forms of LL. In fact, any equivalence relation satisfies a certain minimal form of LL (see below)."

"There are strong and weak versions of RI. The weak version says that RI has some (in fact, many) true instances but also that there are predicates F such that if x and y are the same F, then, for any equivalence relation, E, whatsoever (whether or not an identity relation), E(x,y). This last condition implies that the relation x and y are the same F satisfies LL. The relation P defined so that P(x,y) if and only if H(x) and H(y), where H is some predicate, is an equivalence relation. Hence, if H holds of x but not of y, there is an equivalence relation (namely, P(x,y)) that fails to hold of x and y. If we add that in this instance ‘x and y are the same F’ is to be interpreted in terms of the relation I(A,x,y), then the weak version of RI says that there is such a thing as relative identity and such a thing as absolute identity as well. The strong version, by contrast, says that there are (many) true instances of RI but there is no such thing as absolute identity. It is difficult to know what to make of the latter claim. Taken literally, it is false. The notion of unrestricted identity (in the sense of ‘unrestricted’ explained in §1) is demonstrably coherent. We return to this matter in §5 ."

"The puzzles about identity outlined in §2 (and there are many others, as well as many variants of these) put considerable pressure on the standard account. A theory of identity that allows for instances of RI is an attractive alternative (see below §4). But there is a certain kind of example of RI, frequently discussed in the literature, that has given relative identity something of a bad name. The passenger/person example is a case in point. The noun ‘passenger’ is derived from the corresponding relational expression ‘passenger in (on) …’. A passenger is someone who is a passenger in some vehicle (on some flight, etc.). Similarly, a father is man who fathers someone or who is the father of someone. This way of defining a kind of things from a relation between things is perfectly legitimate and altogether open-ended. Given any relation R, we can define ‘an R’ to apply to anything x that stands in R to something y. For example, we can define a ‘schmapple’ to be an apple in a barrel. All this is fine. But we can't infer from such a definition that the same apple might be two different schmapples. From the fact that someone is the father of two different children, we don't judge that he is two different fathers. The fact that airlines choose to count passengers as they do, rather than track persons, is their business, not logic's."

"However, when R is an equivalence relation, we are entitled to such an inference. Consider the notorious case of "surmen" (Geach 1967). A pair of men are the "same surman" if they have the same surname; and a surman is a man who bears this relation to someone. So now it appears that that two different men can be the same surman, since two different men can have the same surname. As Geach (1967) insists (also Geach 1973), surmen are defined to be men, so they are not merely classes of men. Hence we seem to have an instance of RI, and obviously any similarity relation (e.g., x and y have the same shape) will give rise to a similar case. Yet such instances of RI are not very interesting. It is granted all around that when ‘F’ is adjectival, different Gs may be the same F. Different men may have the same surname, different objects, the same color, etc. Turning an adjectival similarity relation into a substantival one having the form of an identity statement yields an identity statement in name only."

"A word about the point of view of those who subscribe to the weak version of RI. The view (call it the ‘weak view’) is that ordinary identity relations concerning (largely) the world of contingency and change are equivalence relations answering to restricted forms of LL. The exact nature of the restriction depends on the equivalence relation itself, though there is an element of generality. The kinds of properties preserved by the same dog relation are intuitively the same kinds of properties as are preserved by the same cat relation. From a logical point of view the best that can be said is that any identity relation, like any equivalence relation, preserves a certain minimal set of properties. For suppose E is some equivalence relation. Let S be the set containing all formulas of the form E(x,y), and closed under the formation of negations, conjunctions, and quantification. Then E preserves any property expressed by a formula in S. Furthermore, on this view, although absolutely distinct objects may be the same F, absolutely identical objects cannot differ at all. Any instance of RI implies that
x and y are absolutely distinct."

"Let us have a look back at the paradoxes of identity from the perspective of the weak view regarding relative identity. That view allows that absolutely distinct objects may be the same F, but denies that absolutely identical objects can be different G's. This implies that if x and y are relatively different objects, then x and y are absolutely distinct, and hence only pairs of absolutely distinct objects can satisfy RI. If x and y are absolutely distinct, we shall say that x and y are distinct ‘logical objects’; and similarly, if x and y are absolutely identical objects, then x and y are identical logical objects. The term ‘logical object’ does not stand for some new and special kind of thing. Absolutely distinct apples, for example, are distinct logical objects."

"The following is the barest sketch of relativist solutions to the paradoxes of identity discussed in §2. No attempt is made to fully justify any proposed solution, though a modicum of justification emerges in the course of §6. It should be kept in mind that some of the strength of the relativist solutions derives from the weaknesses of the absolutist alternatives, some of which are discussed in §2."

"The alleged Paradox of Change. Young Oscar and old Oscar are the same dog but absolutely different things, i.e. different logical objects. The material conditions rendering young Oscar and old Oscar the same dog (and the same Dalmatian) are precisely the same as the material conditions under which young Oscar and old Oscar would qualify as temporal parts of the same dog. The only difference is logical. The identity relation between young Oscar and old Oscar can be formalized in an extensional logic (Deutsch 1997), but a theory of temporal parts requires a modal/temporal apparatus. Young Oscar is wholly present during his youth and possesses the simple, non-relational, property of not having a gray muzzle."

"The alleged Chrysippus' Paradox. Oscar and Oscar-minus both survive Oscar's loss of a tail. At both t and t′ Oscar and Oscar-minus are the same dog, but at t, Oscar and Oscar-minus are distinct logical objects. This implies (by ND) that Oscar and Oscar-minus are distinct logical objects even at t′ Hence, we must allow that distinct logical objects may occupy the same space at the same time. This is not a problem, however. For although Oscar and Oscar-minus are distinct logical objects at t′, they are physically coincident."

"The alleged Paradox of 101 Dalmatians. The relativist denies that dogs are "maximal." It is not true that no proper part of a dog is dog. All the 101 (and more) proper parts of Oscar differing from him and from one another by a hair are dogs. In fact, many (though of course not all) identity preserving changes Oscar might undergo correspond directly to proper parts of (an unchanged) Oscar. But there is no problem about barking in unison, and no problem about acting independently. All 101 are the same dog, despite their differences, just as young Oscar and old Oscar are the same dog, . The relativist denies that the dogs are many rather than deny that the many are dogs (Lewis 1993)."

"The alleged Paradox of Constitution. Constitution is identity, absolute identity. The relation between the piece of clay c and the statue s1 on day 1 is one of absolute identity. So we have that c = s1 on day 1, and for the same reason, c = s2 on day 2. Furthermore, since s1 and s2 are different statues, it follows (on the weak view) that s1≠s2. In addition, the piece of clay c constituting s1 on day 1 is (relatively) the same piece of clay as the piece of clay constituting s2 on day 2. (The identity is relative because we have distinct objects — the two statues — that are the same piece of clay.) It follows that no name of the piece of clay c can be a rigid designator in the standard sense. That is, no name of c denotes absolutely the same thing on day 1 and on day 2. For on day 1, a name of the piece of clay c would denote s1 and on day 2, it would denote s2, and s1 and s2 are absolutely distinct. Nevertheless, a name of the piece of clay may be relatively rigid: it may denote at each time the same piece of clay. Although no name of the piece of clay c is absolutely rigid, that does not prevent the introduction of a name of c that denotes c at any time (or possible world). (Kraut 1980 discusses a related notion of relative rigidity.)"

"There is, however, a certain ambiguity in the notion of a name of the piece of clay, inasmuch as the piece of clay may be any number of absolutely distinct objects. The notion of relative rigidity presupposes that a name for the piece of clay refers, with respect to some parameter p, to whatever object counts as the piece of clay relative to that parameter. This may be sufficient in the case of the piece of clay, but in other cases it is not. With respect to a fixed parameter p there may be no unique object to serve as the referent of the name. For example, if any number of dog parts count, at a fixed time, as the same dog, then which of these objects serves as the referent of ‘Oscar’? We shall leave this question open for the time being but suggest that it may be worthwhile to view names such as ‘Oscar’ as instantial terms — terms introduced into discourse by means of existential instantiation. The name ‘Oscar’ might be taken as denoting a representative member of the equivalence class of distinct objects qualifiying as the same dog as Oscar. It would follow, then, that most ordinary names are instantial terms. (An alternative is that of Geach 1980, who draws a distinction between a name of and a name for an object; see Noonan 1997 for discussion of Geach's distinction.)"

"The alleged paradox of the Ship of Theseus. In this case, the relativist, as so far understood, may seem to enjoy no advantage over the absolutist. The problem is not clearly one of reconciling LL with ordinary judgments of identity, and the advantage afforded by RI does not seem applicable. Griffin (1977), for example, relying on RI, claims that the original and remodeled ship are the same ship but not the same collection of planks, whereas the reassembled ship is the same collection of planks as the original but not the same ship. This simply doesn't resolve the problem. The problem is that the reassembled and remodeled ships have, prima facie, equal claim to be the original and so the bald claims that the reassembled ship is not—and the remodeled ship is—the original are unsupported. The problem is that of reconciling the intuition that certain small changes (replacement of a single part or small portion) preserve identity, with the problem illustrated by the sandals example of §2.5. It turns out, nevertheless, that the problem is one of dealing with the excesses of LL. To resolve the problem, we need an additional level of relativity. To motivate this development, consider the following abstract counterpart of the sandals example."

"On the left there is an object P composed of three parts, P1, P2, and P3. On the right is an exactly similar but non-identical object, Q, composed of exactly similar parts, Q1, Q2, and Q3, in exactly the same arrangement. For the sake of illustration, we adopt the rule that only replacement of (at most) a single part by an exactly similar part preserves identity. Suppose we now interchange the parts of P and Q. We begin by replacing P1 by Q1 in P and replacing Q1 by P1 in Q, to obtain objects P1 and Q1. So P1 is composed of parts Q1, P2, and P3, and Q1 is composed of parts P1, Q2, and Q3. We then replace P2 in P1 by Q2, to obtain P2, and so on. Given our sample criterion of identity, and assuming the transitivity of identity, P and P3 are counted the same, as are Q and Q3. But this appears to be entirely the wrong result. Intuitively, P and Q3 are the same, as are Q and P3. For P and Q3 are composed of exactly the same parts put together in exactly the same way, and similarly for Q and P3. Futhermore, Q3 (P3) can be viewed as simply the result of taking P (Q) apart and putting it back together in a slightly different location. And this last difference can be eliminated by switching the locations of P3 and Q3 as a last step in the process."

"Suppose, however, that we replace our criterion of identity by the following more complicated rule: x and y are the same relative to z, if both x and y differ from z at most by a single part. (This relation is transitive, and is in fact an equivalence relation.) For example, relative to P, P, P1, Q2, and Q3 are the same, but Q, Q1, P2 and P3, are not. Of course, replacement by a single part is an artificial criterion of identity. In actual cases, it will be a matter of the degree or kind of deviation from the original (represented by the third parameter, z). The basic idea is that identity through change is not a matter of identity through successive, accumulated changes — that notion conflicts with both intuition (e.g., the sandals example) and the Kripkean argument: Through successive changes objects can evolve into other objects. The three-place relation of idenitity does not satisfy LL and is consistent with the outlook of the relativist. Gupta (1980) develops a somewhat similar idea in detail. Williamson (1990) suggests a rather different approach, but one that, like the above, treats identity through change as an equivalence relation that does not satisfy LL."

"Alonzo Church's alleged Paradox. Church's argument implies that if Pierre's doxastic position is as described (in §2.6), then London and Londres are distinct objects. Assuming the standard account of identity, the result is that either Pierre's doxastic position cannot be as described or else London and Londres are different cities (or else we must punt). Since London and Londres are not different cities, the standard account entails that Pierre's doxastic position cannot be as described (or else we must punt). This was Church's own position as regards certain puzzles about synonymy, such as Mate's puzzle (Mates 1952). Church held that one who believes that lawyers are lawyers, must indeed believe that lawyers are attorneys, despite any refusal to assent to (or desire to dissent from) ‘Lawyers are attorneys’ (Church 1954). Kripke later argued (Kripke 1979) that assent and failure to assent must be taken at face value (at least in the case of Pierre) and Pierre's doxastic position is as described. Kripke chose to punt — concluding that the problem is a problem for any "logic" of belief. The relativist concludes instead that (a) Pierre's doxastic position is as described, (b) if so, London and Londres are distinct objects, and (c) London and Londres are nonetheless the same city. Whether this resolution of Church's paradox can be exploited to yield solutions to Frege's puzzle (Salmon 1986) or Kripke's puzzle (1979) remains to be seen. Crimmins (1998) has recently suggested that the analysis of propositional attitudes requires a notion of "semantic pretense." In reporting Pierre's doxastic position we engage in a pretense to the effect that London and Londres are different cities associated with different Fregean senses. Crimmins' goal is to reconcile (a), (c) and the following, (d): that the pure semantics of proper names (’London’, ‘Londres’) is Millian or directly referential (Kripke 1979). The relativist proposes just such a reconciliation but suggests that the pretense can be dropped."

"Objections and Replies. The following constitute a "start up" set of objections and replies concerning relative identity and/or aspects of the foregoing account of relative identity and its rival. Time and space constraints prevent a more extended initial discussion. In addition, there is no presumption that the objections discussed below are the most important or that the initial replies to them are without fault. It is hoped that the present discussion will evolve into a more full blown one, involving contributions by the author and readers alike. Should the discussion become lengthy, old or unchallenged objections and/or replies can be placed in the archives."

"Objection 1: "Relativist theories of identity, all of which are inconsistent with Leibniz's principle [LL], currently enjoy little support. The doubts about them are (a) whether they really are theories of numerical identity, (b) whether they can be made internally consistent, and (c) whether they are sufficiently motivated." (Burke 1994.)"

"Reply: In reverse order: (c) The issues discussed in §2 and §4 surely provide sufficient motivation. (b) No proof of inconsistency has ever been forthcoming from opponents of relative identity, and in fact the weak view is consistent inasmuch as it has a model in the theory of similarity relations. The arguments outlined in the second paragraph of §3 are frequently cited as showing that relative identity is incoherent; but they show only that RI is incompatible with (unrestricted) LL. (a) See the replies to objections 2 and 3 below."

"Objection 2: If an identity relation obeys only a restricted form of LL — if it preserves only some properties and not all — then how do we tell which properties serve to individuate a pair of distinct objects?"

"Reply: Similarity relations satisfy only restricted forms of LL. How then do we tell which properties are preserved by the same shape relation and which are not? It is no objection to the thesis that identity relations in general preserve some properties and not others to demand to know which are which. At best the objection points to a problem we must face anyway (for the case of similarity). In general, a property is preserved by an equivalence relation if it "spreads" in an equivalence class determined by the relation: If one member of the class has the property, then every member does. Every property spreads in a singleton, as absolute identity demands."

"Objection 3: If identity statements are mere equivalencies, what distinguishes identity from mere similarity?"

"Reply: The distinction between identity and similarity statements (or sentences) is usually drawn in terms of the distinction between substantival and adjectival common nouns. If F is a common noun standing for a kind of things e.g., ‘horse’, then ‘x and y are the same F’ is a statement of identity, whereas if F is an a common noun standing for a kind of properties of things, then ‘x and y are the same F’ is a statement of similarity. (It's interesting to note that when the noun is proper, i.e. a proper name, the result is a statement of similarity, not identity — as in ‘He's not the same Bill we knew before’.) This distinction rests ultimately on the metaphysical distinction between substance and attribute, object and property. While the distinction no doubt presupposes the concept of individuation (the bundle theory, for example, presupposes that we have the means to individuate properties), there is no obvious reason to suppose that it entails the denial of RI, i.e. the claim that no instance of RI is true. For a beginner's review — from an historical perspective — of the issues concerning substance and attribute, see O'Connor, (1967); and for more recent and advanced discussion and bibliography, see the entry on properties."

"Objection 4: Consider the following alleged instance of RI:"

"1.A is the same word type as B, but A and B are different word tokens.
"If ‘A’ and ‘B’ refer to the same objects throughout (1), the first conjunct of (1) is not an identity statement, and the counterexample (to the thesis that no instance of RI is true) fails. If both conjuncts are identity statements in the required sense, ‘A’ and ‘B’ must refer to word types in the first conjunct and word tokens in the second, and the counterexample fails" (Perry 1970)."

"Reply: First, if "in the required sense" means "satisfies LL," then the objection buys correctness only at the price of begging the question. Advocates of relative identity will maintain that the relation A is the same word type as B is an identity relation, defined on tokens, that does not satisfy LL."

"Secondly, even if one insists that in this case intuition dictates that if A and B refer to tokens in both conjuncts of (1), then ‘A is the same word type as B’ expresses only the similarity relation: A and B are tokens of the same type, there are other cases where, intuitively, both conjuncts of RI involve identity relations and yet the relevant terms all refer to the same kind of things; for example,"

"2.A and B are the same dog but A and B are different physical objects,
as said of young Oscar and old Oscar. Here there is no temptation to suppose that the relation A and B are the same dog is not an identity relation. One may invoke a theory — a theory of temporal parts, for example — that construes the relation as a certain kind of similarity, but that is theory, not pretheoretical intuition. It is no objection to the relativist's theory, which holds in part that ‘A and B are the same dog’ expresses a relation of primitive identity, that there is an alternative theory according to which it expresses a similarity relation obtaining between two temporal parts of the same object. Furthermore, in the case of (2), A and B refer, again intuitively, to the same things in both conjuncts."

"Third, there are cases in which the relative identity view does possess an ontological advantage. Consider"

3.A and B are the same piece of clay but A and B are different statues.
Suppose A and B are understood to refer to one sort of thing — pieces of clay — in the first conjunct and another — statues — in the second conjunct. Assume that the piece of clay c denoted by A in the first conjunct constitutes, at time t, the statue s. Then assuming that statues are physical objects, there are two distinct physical objects belonging to different kinds occupying the same space at t. Some, notably Wiggins (1980), hold that this is entirely possible: Distinct physical objects may occupy the same space at the same time, provided they belong to different kinds. The temporal parts doctrine supports and encourages this view. A statue may be a temporal part of a temporally extended piece of clay. But one statue, it seems, cannot be a temporal part of another. Even so, however, the duality of constituter and thing constituted is unparsimonious (cf. Lewis 1993), and the relativist is not committed to it."

Again, consider

4.A and B are the same book but A and B are different copies (of the book).
One can say that in the first conjunct, A and B refer to books (absolutely the same book), whereas in the second conjunct, A and B refer to (absolutely distinct) copies. But the alleged duality of books and copies of books is unparsimonious and the relativist is not committed to it. There is no reason to concede to the philosopher that we do not actually purchase or read books; instead we purchase and read only copies of books. Any copy of a book is just as much the "book itself" as is any other copy. Any copy of a book is the same book as any other copy. Nelson Goodman once remarked that "Any accurate copy of a poem is as much the original work as any other" (Goodman 1968). Goodman was not suggesting that the distinction between poem and copy collapses. If it does collapse, however, we have an explanation of why any accurate copy is as much the original work as any other: any such copy is the same work as any other."

"Objection 5: Geach remarks that "As for our recognizing relative identity predicables: any equivalence relation…can be used to specify a criterion of relative identity." But §3 above contains a counterexample. Some equivalence relations are defined in terms of the I-predicable of a theory and hence cannot serve as such. (Any pair of I-predicables for a fixed theory are equivalent.) In fact it seems that any equivalence relation presupposes identity (cf. McGinn 2000). For example, the relation x and y are the same color presupposes identity of colors, since it means that there are colors C and C′ such that x has C and y has C′, and C = C′. Identity, therefore, is logically prior to equivalence."

"Reply: This is a good objection. It does seem to show, as the objector says, that identity is logically prior to ordinary similarity relations. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as x and y are the same color have been represented, in the way indicated in the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. In Deutsch (1997), an attempt is made to treat similarity relations of the form ‘x and y are the same F’ (where F is adjectival) as primitive, first-order, purely logical relations (see also Williamson 1988). If successful, a first-order treatment of similarity would show that the impression that identity is prior to equivalence is merely a misimpression — due to the assumption that the usual higher-order account of similarity relations is the only option."

"Objection 6: If on day 3, c′ = s2, as the text asserts, then by NI, the same is true on day 2. But the text also asserts that on day 2, c = s2; yet c ≠ c′. This is incoherent."

"Reply: The term s2 is not an absolutely rigid designator and so NI does not apply."

"Objection 7: The notion of relative identity is incoherent: "If a cat and one of its proper parts are one and the same cat, what is the mass of that one cat?" (Burke 1994)"

"Reply: Young Oscar and Old Oscar are the same dog, but it makes no sense to ask: "What is the mass of that one dog." Given the possibility of change, identical objects may differ in mass. On the relative identity account, that means that distinct logical objects that are the same F may differ in mass — and may differ with respect to a host of other properties as well. Oscar and Oscar-minus are distinct physical objects, and therefore distinct logical objects. Distinct physical objects may differ in mass."

"Objection 8: We can solve the paradox of 101 Dalmatians by appeal to a notion of "almost identity" (Lewis 1993). We can admit, in light of the "problem of the many" (Unger 1980), that the 101 dog parts are dogs, but we can also affirm that the 101 dogs are not many; for they are "almost one." Almost-identity is not a relation of indiscernibility, since it is not transitive, and so it differs from relative identity. It is a matter of negligible difference. A series of negligible differences can add up to one that is not negligible."

"Reply: The difference between Oscar and Oscar-minus is not negligible and the two are not almost-identical. Lewis concedes this point but proposes to combine almost-identity with supervaluations to give a mixed solution to the paradox. The supervaluation solution starts from the assumption that one and only one of the dog parts is a dog (and a Dalmatian, and Oscar), but it doesn't matter which. It doesn't matter which because we haven't decided as much, and we aren't going to. Since it is true that any such decision renders one and only one dog part a dog, it is plain-true, i.e. supertrue, that there is one and only one dog in the picture. But it is not clear that this approach enjoys any advantage over that of relative identity; in fact, it seems to produce instances of RI. Compare: Fred's bicycle has a basket attached to it. Ordinarily, our discourse slides over the difference between Fred's bicycle with its basket attached and Fred's bicycle minus the basket. (In this respect, the case of Fred's bicycle differs somewhat from that of Oscar and Oscar-minus. We tend not to ignore that difference.) In particular, we don't say that Fred has two bicycles even if we allow that Fred's bicycle-minus is a bicycle. Both relative identity and supervaluations validate this intuition. However, both relative identity and supervaluations also affirm that Fred's bicycle and Fred's bicycle-minus are absolutely distinct objects. That is, the statement that Fred's bicycle and Fred's bicycle-minus are distinct is supertrue. So the supervaluation technique affirms both that Fred's bicycle and Fred's bicycle-minus are distinct objects and that there is one and only one (relevant) bicycle. That is RI, or close enough. The supervaluation approach is not so much an alternative to relative identity as a form of it."