Identity is often said to be a relation each thing bears to itself andto no other thing (e.g., Zalabardo 2000). This characterization isclearly circular (“noother thing”) andparadoxical too, unless the notion of “each thing” isqualified. More satisfactory (though partial) characterizations areavailable and the idea that such a relation of absolute identityexists is commonplace. Some, however, deny that a relation of absoluteidentity exists. Identity, they say, is relative: It is possible forobjects \(x\) and \(y\) to be the same \(F\) and yetnot thesame \(G\) (where \(F\) and \(G\) are predicates representing kinds ofthings—apples, ships, passengers—rather than merelyproperties of things—colors, shapes). In such a case“same” cannot mean absolute identity. For example, thesame person might be two different passengers, since one person may becounted twice as a passenger. If to say that \(x\) and \(y\) are thesame person is to say that \(x\) and \(y\) are persons and are(absolutely) identical, and to say that \(x\) and \(y\) are differentpassengers is to say that \(x\) and \(y\) are passengers and are(absolutely) distinct, we have a contradiction. Others maintain thatwhile there are such cases of “relative identity”, thereis also such a thing as absolute identity. According to this view,identity comes in two forms: trivial or absolute and nontrivial orrelative (Gupta 1980). These maverick views present a seriouschallenge to the received, absolutist doctrine of identity. In thefirst place, cases such as the passenger/person case are moredifficult to dismiss than might be supposed (but see below, §3).Secondly, the standard view of identity is troubled by many persistentpuzzles and problems, some of recent and some of ancient origin. Therelative identity alternative sheds considerable light on theseproblems even if it does not promise a resolution of them all.

A word about notation. In what follows, lower-case italic letters“\(x\)”, “\(y\)”, etc., are used informallyeither as variables (bound or free) or as (place-holders for)individual constants. The context should make clear which usage is inplay. Occasionally, for emphasis or in deference to logical tradition,other expressions for individual constants are employed. Also, theuse/mention distinction is not strictly observed; but again thecontext should resolve any ambiguity.

• 1. The Standard Account of Identity
• 2. Paradoxes of Identity
  • 2.1 The Paradox of Change
  • 2.2 Chrysippus’ Paradox
  • 2.3 The Paradox of 101 Dalmatians
  • 2.4 The Paradox of Constitution
  • 2.5 The Ship of Theseus Paradox
  • 2.6 Church’s Paradox
  • 2.7 The Theological Paradox
• 3. Relative Identity
• 4. The Paradoxes Reconsidered
  • 4.1 The Paradox of Change
  • 4.2 Chrysippus’ Paradox
  • 4.3 The Paradox of 101 Dalmatians
  • 4.4 The Paradox of Constitution
  • 4.5 The Ship of Theseus Paradox
  • 4.6 Church’s Paradox
  • 4.7 The Theological Paradox
• 5. Absolute Identity
• 6. Objections and Replies
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Standard Account of Identity

[Note: The following material is somewhat technical. The reader maywish to casually review it now and return to it as needed, especiallyin connection with §5. The propositions \(\text{Ref},\text{LL}, \text{Ref}', \text{LL}', \text{NI}\), and\(\text{ND}\) are identified in the present section and are referredto as such in the rest of the entry.]

Identity may be formalized in the language \(L\) of classicalfirst-order logic (FOL) by selecting a two-place predicate of \(L\),rewriting it as “=”, and adopting the universal closuresof the following two postulates:

where the formula \(\phi(x)\) is like the formula \(\phi(y)\) exceptfor having occurrences of \(x\) at some or all of the places\(\phi(y)\) has occurrences of \(y\) (see Enderton 2000, for a precisedefinition). Ref is the principle of thereflexivity ofidentity and LL (Leibniz’ Law) is the principle oftheindiscernibility of identicals. It says in effect thatidentical objects cannot differ in any respect. The othercharacteristic properties of identity,symmetry \((x = y\rightarrow y = x)\), andtransitivity \((x = y \amp y = z\rightarrow x = z)\), may be deduced from Ref and LL. Any relationthat is reflexive, transitive, and symmetric is called an“equivalence relation”. Thus, identity is an equivalencerelation satisfying LL. But not all equivalence relations satisfy LL.For example, the relationx and y are the same size is anequivalence relation that does not satisfy LL (with respect to a richlanguage such as English).

Let \(E\) be an equivalence relation defined on a set \(A\). For \(x\)in \(A\), \([x]\) is the set of all \(y\) in \(A\) such that \(E(x,y)\); this isthe equivalence class of x determined by E. Theequivalence relation \(E\) divides the set \(A\) into mutuallyexclusive equivalence classes whose union is \(A\). The family of suchequivalence classes is called “the partition of \(A\) induced by\(E\)”.

Now let \(A\) be a set and define the relation \(I(A,x,y)\) asfollows: For \(x\) and \(y\) in \(A\), \(I(A,x,y)\) if and only if foreach subset \(X\) of \(A\), either \(x\) and \(y\) are both elementsof \(X\) or neither is an element of \(X\). This definition isequivalent to the more usual one identifying the identity relation ona set \(A\) with the set of ordered pairs of the form \(\langlex,x\rangle\) for \(x\) in \(A\). The present definition proves morehelpful in what follows.

Suppose for the moment that we do not assign any specialinterpretation to the identity symbol. We treat it like any othertwo-place predicate. Let \(M\) be a structure for \(L\) and assumethat Ref and LL are true in \(M\). Call the relation defined in \(M\)by the conjunction of Ref and LL “indiscernibility” (seeEnderton 2000, for the definition of definability in a structure).There are three important points to note about the relationshipbetween indiscernibility, and the relation \(I(A,x,y)\). First,indiscernibility need not be the relation \(I(A,x,y)\) (where \(A\) isthe domain of the structure). It might be an equivalence relation\(E\) having the property that for some elements \(u,v\), of thedomain, \(E(u,v)\) holds, although \(I(A,u,v)\) fails. Secondly, thereis no way to “correct for” this possibility. There is nosentence or set of sentences that could be added to the list beginningwith Ref and LL that would guarantee that indiscernibility coincideswith \(I(A,x,y)\). This fact is usually expressed by saying thatidentity is not a first-order or “elementary” relation.(For a proof, see Hodges 1983.) However, in a language such as settheory (as usually interpreted) or second-order logic, in which thereis a quantifier “all \(X\)” permitting quantification overall subsets of a given set, \(I(A,x,y)\) is definable (Freund2019).

Third, given any structure \(M\) for \(L\) in which Ref and LL aretrue, there is a corresponding structure \(\textit{QM}\), the“quotient structure” determined by \(M\), in whichindiscernibilitydoes coincide with \(I(A,x,y)\).\(\textit{QM}\) is obtained in roughly the following way: Let theelements of \(\textit{QM}\) be the equivalence classes \([x]\), forelements \(x\) of \(M\) determined by indiscernibility in \(M\). If\(F\) is a one-place predicate true in \(M\) of some object \(x\) in\(M\), then define \(F\) to be true of \([x]\) in \(\textit{QM}\), andsimilarly for many-place predicates and constants. It can then beshown that any sentence true in \(M\) is true in \(\textit{QM}\), andvice versa. The existence of quotient structures makes it possible totreat the identity symbol as a logical constant interpreted in termsof \(I(A,x,y)\). There is in fact in general no other way toguarantee that Ref and LL will hold in every structure. (AsQuine [1970] points out, however, afinite language willalways contain a predicate satisfying Ref and LL in any structure; cf.Hodges 1983.) The alternative, however, is to view FOL with Ref and LL(FOL⁼) as a proper theory in whose models (structures inwhich Ref and LL hold) there will be an equivalence relation \(E\)such that if \(E(x,y)\) holds, then \(x\) and \(y\) will beindiscernible with respect to thedefined subsets of thedomain. But we cannot in general assume that every subset of thedomain is definable. If the domain is infinite, \(L\) runs out ofdefining formulas long before the domain runs out of subsets.Nonetheless, a strong metatheorem asserts that any set of formulasthat has a model, has a countable (finite or denumerable) model. Thismeans that the difference between indiscernibility and \(I(A,x,y)\) isminimized at least to the extent that, for a sufficiently richlanguage such as \(L\), the valid formulas concerning indiscernibility(i.e., the formulas true in every model of what is termed below“the pure \(L\)-theory with identity”) coincide with thevalid formulas concerning \(I(A,x,y)\). (See Epstein 2001 for a sketchof a proof of this fact.) This is not to say, however, that thereisn’t a significant difference between identityquaindiscernibility and identityqua \(I(A,x,y)\) (see below).Both points of view—that FOL⁼ is a proper theory andthat it is a logic—may be found in the literature (Quine 1970).The latter is the more usual view and it will count here as part ofthe standard account of identity.

Assume that \(L'\) is some fragment of \(L\) containing a subset ofthe predicate symbols of \(L\) and the identity symbol. Let \(M\) be astructure for \(L'\) and suppose that some identity statement \(a =b\) (where \(a\) and \(b\) are individual constants) is true in \(M\),and that Ref and LL are true in \(M\). Now expand \(M\) to a structure\(M'\) for a richer language—perhaps \(L\) itself. That is,assume we add some predicates to \(L'\) and interpret them as usual in\(M\) to obtain an expansion \(M'\) of \(M\). Assume that Ref and LLare true in \(M'\) and that the interpretation of the terms \(a\) and\(b\) remains the same. Is \(a = b\) true in \(M'\)? That depends. Ifthe identity symbol is treated as a logical constant, the answer is“yes”. But if it is treated as a non-logical symbol, thenit can happen that \(a = b\) is false in \(M'\). The indiscernibilityrelation defined by the identity symbol in \(M\) may differ from theone it defines in \(M'\); and in particular, the latter may be more“fine-grained” than the former. In this sense, if identityis treated as a logical constant, identity isnot“language relative”; whereas if identity is treated as anon-logical notion, it \(is\) language relative. For this reason wecan say that, treated as a logical constant, identity is“unrestricted”. For example, let \(L'\) be a fragment of\(L\) containing only the identity symbol and a single one-placepredicate symbol; and suppose that the identity symbol is treated asnon-logical. The formula

is then true in any structure for \(L'\) in which Ref and LL are true.The reason is that the unique one-place predicate of \(L'\) dividesthe domain of a structure into those objects it satisfies and those itdoes not. Hence, at least two of any group of three objects will beindiscernible. On the other hand, if the identity symbol isinterpreted as \(I(A,x,y)\), this formula is false in any structurefor \(L'\) with three or more elements.

If we do wish to view identity as a non-logical notion, then thephenomenon of language relativity suggests that it is best not toformalize identity using a single identity predicate “=”.Instead, we have the following picture: We begin with a language \(L\)and define anL-theory with identity to be a theory whoselogical axioms are those of FOL and which is such that \(L\) containsa two-place predicate \(E_L\) satisfying the non-logical axiom\(\textrm{Ref}\,'\) and the non-logical axiom schema\(\textrm{LL}'\)

Thepure \(L\)-theory with identity is the \(L\)-theory whosesole non-logical axiom is \(\textrm{Ref}\,'\) and whose solenon-logical axiom schema is \(\textrm{LL}'\).

Now the phenomenon of language relativity can be described moreaccurately as follows. Let \(L_1\) be a sublanguage of \(L_2\) andassume that \(T_1\) and \(T_2\) are, respectively, the pure\(L_1\)-theory with identity and the pure \(L_2\)-theory withidentity. Let \(M_1\) and \(M_2\) be models of \(T_1\) and \(T_2\),respectively, having the same domain. Assume that \(a\) and \(b\) areindividual constants having the same interpretation in \(M_1\) and\(M_2\). Let \(E_1\) and \(E_2\) be the identity symbols of \(L_1\)and \(L_2\). It can happen that \(E_1 (a,b)\) is true in \(M_1\) but\(E_2 (a,b)\) is false in \(M_2\). We can then say, with Geach (1967;see §4) and others, that the self-same objects indiscernibleaccording to one theory may be discernible according to another.

There are two further philosophically significant features of thestandard account of identity. First, identity is anecessaryrelation: If \(a\) and \(b\) are rigid terms (terms whose referencedoes not vary with respect to parameters such as time or possibleworld) then

(NI)

If \(a = b\) is true, then it is necessarily true.

Assuming certain modal principles, the necessity of distinctness (ND)follows from NI.

(ND)

If \(a \ne b\) is true, then it is necessarily true.

Note that the necessary truth of \(a = b\) does not imply thenecessary existence of objects \(a\) or \(b\). We may assume that whata rigid term \(a\) denotes at a possible world (or moment of time)\(w\) need not exist in \(w\). Secondly, we do not ordinarily saythings of the form “\(x\) is the same as \(y\)”. Instead,we say “\(x\) and \(y\) are the same person” or“\(x\) and \(y\) are the same book”. The standard view isthat the identity component of such statements is just “\(x\) isthe same as \(y\)”. For example, according to the standard view,“\(x\) and \(y\) are the same person” reduces to“\(x\) and \(y\) are persons and \(x\) is the same as\(y\)”,where the second conjunct may beformalized as in FOL⁼.

2. Paradoxes of Identity

The concept of identity, simple and settled though it may seem (ascharacterized by the standard account), gives rise to a great deal ofphilosophical perplexity. A few (by no means all) of the salientproblems are outlined below. These are presented in the form ofparadoxes—arguments from apparently undeniable premises toobviously unacceptable conclusions. The aim here is to make clear justwhat options are available to one who would stick close to thestandard account. Often (but not always) little or no defense orcritique of any particular option is offered. In the next section, weshall see what the relative identity alternative offers bycomparison.

2.1 The Paradox of Change

The most fundamental puzzle about identity is the problem of change.Suppose we have two photographs of a dog, Oscar. In one, \(A\), Oscaris a puppy, in the other, \(B\), he is old and gray-muzzled. Yet wehold that he is the same dog, in, it appears, direct violation of LL.More explicitly, \(B\) is a photograph of an old dog with a graymuzzle; \(A\) is a photograph of a young dog without a gray muzzle.\(A\) and \(B\) are photographs of the same dog. But according to LL,if the dog in \(B\) has a property (e.g., having a gray muzzle) thatthe dog in \(A\) lacks, then \(A\) and \(B\) arenotphotographs of the same dog. Contradiction.

Various solutions have been proposed. The most popular are thefollowing two: (1) Simple properties such as having or lacking a graymuzzle are actually relations to times. Oscar has the property oflacking a gray muzzleat time t and the property of having agray muzzle at (a later) \(t'\); but there is no incompatibility,since being thus and so related to time \(t\) and not being thus andso related to time \(t'\) are compatible conditions, and hence changeinvolves no violation of LL. (2) Oscar is an object that is extendedin time as well as space. The puppy Oscar and old gray-muzzled Oscarare distinct temporal parts or stages of the whole temporally extendedOscar. Both photographs are therefore not photographs of the wholeOscar at all because there cannot be still photographs of Oscar.

These proposals may seem plausible, and indeed most philosopherssubscribe to one or other of them. The most commonobjections—that on the temporal parts account, objects are not“wholly present” at any given time, and that on therelations-to-times account, seemingly simply properties of objects,such as Oscar having a gray muzzle, are complicated relations—dolittle more than affirm what their targets deny. Yet the objectionsare an attempt to give voice to a strong intuition concerning ourexperience as creatures existing in time. Both (1) and (2) treat timeand change from a “God’s eye” point of view. (1)presupposes time laid out “all at once”, so to speak, andsimilarly for (2). But we experience no such thing. Instead, while weare prepared to wait to see the whole of a baseball game we arewatching, we are not prepared to wait to see the whole of a paintingwe are viewing.

2.2 Chrysippus’ Paradox

The following paradox—a variation of the paradox ofchange—raises some new questions. It is due to the Stoicphilosopher Chrysippus (c. 280 BCE – c. 206 BCE) and has beenresurrected by Michael Burke (1994). In Chrysippus’s example,“Dion” is the name of a human and “Theon” isthe name of the proper part of Dion that includes everything but his[Dion’s] left foot. Dion then loses his foot and Chrysippuswishes to know which of those objects—Dion orTheon—survives after Dion loses the foot. Burke’s exampleis to suppose that, at some point \(t'\) in the future, poor Oscarwill lose his tail. He then considers the proper part of Oscar, as heis now (at \(t)\), consisting of the whole of Oscar minus his tail.Call this object “Oscar⁻”. According tothe standard account of identity, Oscar and Oscar⁻are distinct at \(t\) and hence, by ND, they are distinct at \(t'\).(Intuitively, Oscar and Oscar⁻ are distinct at \(t'\)since Oscar has a property at \(t'\) that Oscar⁻lacks, namely, the property of having had a tail at \(t\). Notice thatthis argument involves a tacit appeal to ND—or NI, depending onhow you look at it.) Hence, if both survive, we have a case of twodistinct physical objects occupying exactly the same space at the sametime. Assuming that is impossible, and assuming, as commonsensedemands, that Oscar survives the loss of his tail, it follows thatOscar⁻ does not survive. This conclusion isparadoxical because it appears thatnothing happens toOscar⁻ in the interval between \(t\) and \(t'\) thatwould cause it to perish.

The first group of solutions does not take into consideration theamputation event that occurred at \(t'\) when they account for whatexists—you may dub themuntemporalized solutions. Oneextreme option in this category is to deny that there are such thingsas Oscar⁻. Undetached proper parts of objectsdon’t exist (van Inwagen 1981). A less extreme version allowsfor commonsense undetached parts, like tails or feet—of courseOscar⁻ does not belong to this category, so it doesnot exist (Carmichael 2020). On the other hand, a more extreme optionis to deny the existence of both Oscar andOscar⁻—this may follow from some reductionistview like the one that only elementary particles exist. A third, lessextreme, option is to insist that objects ofdifferent kinds,e.g., a clay statue and the piece of clay it is composed of,can occupy the same space at the same time, but objects ofthe same kind, e.g., two statues, cannot (Wiggins 1968; forrefinements, see Oderberg 1996). So Oscar and Oscar⁻are different despite the fact that from \(t'\) onwards they occupythe same space. Guillon (2021) notes that there are three varieties ofthis colocationist view. First, one may maintain that they aredifferent despite sharing all parts: every proper part of Oscar is aproper part of Oscar⁻ andvice versa. Thesecond colocationist variety allows for two different material objectsto be sums of the same set of material objects—so Oscar andOscar⁻ may be different although they are ultimatelycomposed of the same particles. Finally, one of them, most probablyOscar⁻, may be considered a proper part of the other.Obviously, these three options are meaningful only if one doesnot assume the full strength ofgeneral existentional mereology. The last global option is to claim that Oscar andOscar⁻ are two distinct temporally extendedobjects—a dog part, Oscar⁻, and a dog, Oscar,which overlaps at \(t'\). Temporal parts of distinct objects canoccupy the same space at the same time.

The solutions in the second group,temporalized solutions,recognize the world’s ontological inventory has changed at\(t'\). For instance, Chrysippus, according to Philo of Alexandria,claimed that “it is necessary that Dion remains while Theon hasperished” (Long and Sedley 1987, p. 172), so Dion existsthroughout the whole period, but Theon dies at \(t'\). Accepting somekind of mereological essentialism (Chisholm 1973), one may hold anopposite view: since all parts are essential to Oscar, the amputationevent killed Oscar while Oscar⁻ either was created bythe event or enjoyed existence throughout the whole period in question.Then you may maintain that the event at \(t'\) either (i) broughtOscar⁻ into existence or (ii) did not disruptOscar⁻’s prior existence (i.e.,Oscar⁻ existed at all times from \(t\) to \(t'\)).Finally, a proponent ofcontingent identity may claim that att Oscar is different fromOscar⁻, but at \(t'\) they become identical.

2.3 The Paradox of 101 Dalmatians

(This paradox is also known as the paradox of 1001 cats; Geach 1980,Lewis 1993.) Focus on Oscar and Oscar⁻ at\(t\)—before Oscar loses his tail. Is Oscar⁻ adog? When Oscar loses his tail the resulting creature is certainly adog. Why then should we deny that Oscar⁻ is a dog? Wesaw above that one possible response to Chrysippus’ paradox wasto claim that Oscar⁻ does not exist from \(t'\) on.But even if we adopt this view, how does it follow thatOscar⁻, existing as it does at \(t\), is not a dog?Yet if Oscar⁻ is a dog, then, given the standardaccount of identity, there are two dogs where we would normally countonly one. In fact, for each of Oscar’s hairs, of which there areat least 101, there is a proper part of Oscar—Oscar minus ahair—which is just as much a dog as Oscar⁻.There are then at least 101 dogs (and in fact many more) where wewould count only one. Some claim that things such as dogs are“maximal”. No proper part of a dog is a dog (Burke 1993).One might conclude as much simply to avoid multiplying the number ofdogs populating the space reserved for Oscar alone. But the maximalityprinciple may seem to be independently justified as well. When Oscarbarks, do all these different dogs bark in unison? If a thing is adog, shouldn’t it be capable of independent action? YetOscar⁻ cannot act independently of Oscar.Nevertheless, David Lewis (1993) has suggested a reason for countingOscar⁻ and all the 101 dog parts that differ (invarious different ways) from one another and Oscar by a hair, as dogs,and in fact as Dalmatians (Oscar is a Dalmatian). Lewis invokesUnger’s (1980) “problem of the many”. Oscar shedscontinuously but gradually. His hairs loosen and then dislodge, somesuch remaining still in place. Hence, within Oscar’s compass atany given time there are congeries of Dalmatian parts sooner or laterto become definitely Dalmatians; some in a day, some in a second, or asplit second. It seems arbitrary to proclaim a Dalmatian part that isa split second away from becoming definitely a Dalmatian, a Dalmatian,while denying that one a day away is a Dalmatian. As Lewis puts it, wemust either deny that the “many” are Dalmatians, or wemust deny that the Dalmatians are many. Lewis endorses proposals ofboth types but seems to favor one of the latter type according towhich the Dalmatians are not many but rather “almost one”.In any case, the standard account of identity seems unable on its ownto handle the paradox of 101 Dalmatians. It requires that we eitherdeny that Oscar minus a hair is a dog—and a Dalmatian—orelse that we must affirm that there is a multiplicity of Dalmatians,all but one of which is incapable of independent action and all ofwhich bark in unison no more loudly than Oscar barks alone.

2.4 The Paradox of Constitution

Suppose that on day 1 Jones purchases a piece of clay \(c\) andfashions it into a statue \(s_1.\) On day 2, Jones destroys \(s_1\),but not \(c\), by squeezing \(s_1\) into a ball and fashions a newstatue \(s_2\) out of \(c\). On day 3, Jones removes a part of\(s_2\), discards it, and replaces it using a new piece of clay,thereby destroying \(c\) and replacing it by a new piece of clay,\(c'\). Presumably, \(s_2\) survives this change. Now what is therelationship between the pieces of clay and the statues they“constitute”? A natural answer is: identity. On day \(1,c\) is identical to \(s_1\) and on day \(2, c\) is identical to\(s_2\). On day \(3, s_2\) is identical to \(c'\). But this conclusiondirectly contradicts NI. If, on day \(1, c\) is (identical to)\(s_1\), then it follows, given NI, that on day \(2, s_1\) is \(s_2\)(since \(c\) is identical to \(s_2\) on day 2) and hence that \(s_1\)exists on day 2, which it does not. By a similar argument, on day \(3,c\) is \(c'\) (since \(s_2\) is identical to both) and so \(c\) existson day 3, which it does not. We might conclude, then, that eitherconstitution is not identity or that NI is false. Neither conclusionis wholly welcome. Once we adopt the standard account less NI, thelatter principle follows directly from the assumption that individualvariables and constants in quantified modal logic are to be handledexactly as they are in first-order logic. And if constitution is notidentity, and yet statues, as well as pieces of clay, are physicalobjects (and what else would they be?), then we are again forced toaffirm that distinct physical objects may occupy (exactly) the samespace at the same time. The statue \(s_1\) and the piece of clay \(c\)occupy the same space on day 1. Even if this is deemed possible(Wiggins 1980), it is unparsimonious. The standard account is thusprima facie incompatible with the natural idea thatconstitution is identity.

Philosophers have not argued by direct appeal to NI or ND. Typically(e.g., Gibbard 1975, Noonan 1993, Johnston 1992), arguments that \(c\)and \(s_1\) are not identical run as follows: \(c\) exists prior tothe existence of \(s_1\) and hence the two are not identical. Again,\(s_1\) possesses the property of being such that it will be destroyedby being squeezed into a ball, but \(c\) does not possess thisproperty \((c\) will be squeezed into a ball but it will not therebybe destroyed). So again the two are not identical. Further, whateverthe future in fact brings,c might have been squeezed into aball and not destroyed. Since that is not true of \(s_1\), the two arenot identical. On a careful analysis, however, each of these argumentscan be seen to rely on NI or ND,provided one adopts the standardaccount of modal/temporal predicates. This last proviso suggestsan interesting way out for one who adheres to the standard account ofidentity but who also holds that constitution is identity (seebelow).

Some philosophers find it important or at least expedient to frame theissue in terms of the case of a statue \(s\) and piece of clay \(c\)that coincide throughout their entire existence. We bring both \(c\)and \(s\) into existence by joining two other pieces of clay together,or we do something else that guarantees total coincidence. It seemsthat total coincidence is supposed to lend plausibility to the claimthat, in such a case at least, constitution is identity (and hence NIis false—Gibbard 1975). It may do so, psychologically, but notlogically. The same sorts of arguments as against the thesis thatconstitution is identity apply in such a case. For example, \(s\) maybe admired for its aesthetic traits, even long after it ceases toexist, but this need not be true of \(c\). And \(s\) has the property,which \(c\) lacks, of being destroyed if squeezed into a ball. Thosewho defend the thesis that constitution is identity need to defend itin the general case of partial coincidence; and those who attack thethesis do so with arguments that work equal well against both totaland partial coincidence. The assumption that \(s\) and \(c\) aretotally coincident is therefore inessential.

The doctrine of temporal parts offers only limited help. The statementthat \(c\) is identical to \(s_1\) on day 1 but identical to \(s_2\)on day 2 can be construed to mean that \(c\) is a temporally extendedobject whose day 1 stage is identical to \(s_1\) and whose day 2 stageis identical to \(s_2\). Since the two stages are not identical, NIdoes not apply. Similarly, we can regard \(s_2\) as a temporallyextended object that overlaps \(c\) on day 2 and \(c'\) on day 3. Butunless temporal parts theorists are prepared to defend a doctrine ofmodally extended objects—objects extended through possibleworlds analogous to objects extended in time—there remains aproblem. \(s_2\)might have been made of a different piece ofclay, as is in fact the case on day 3. That is, it is logicallypossible for \(s_2\) to fail to coincide with the day 2 stage of\(c\). But it is not logically possible for the day 2 stage of \(c\)to fail to coincide with itself.

Lewis recognizes this difficulty and proposes to deal with it byappealing to his counterpart theory (Lewis 1971, 1986, and 1993).Different concepts, e.g.,statue andpiece of clay,are associated with different counterpart relations and hence withdifferent criteria of trans-world identity. This has the effect ofrendering modal predicates “Abelardian” (Noonan 1991,1993). The property determined by a modal predicate may be affected bythe subject term of a sentence containing the predicate. The subjectterm denotes an object belonging to this or that kind or sort. Butdifferent kinds or sorts may determine different properties (ordifferent counterpart relations). In particular, the propertiesdetermined by the predicate “might not have coincided with\(c_2\)” (where \(c_2\) names the day 2 stage of \(c)\) in thefollowing sentences,

1. \(s_2\) might not have coincided with \(c_2\),
2. \(c_2\) might not have coincided with \(c_2\),

aredifferent, and hence (a) and (b) are compatible, evenassuming that \(s_2\) and \(c_2\) are identical. (It should beemphasized that counterpart theory is not the only means of obtainingAbelardian predicates. See Noonan 1991.)

The upshot seems to be that the advocate of the standard account ofidentity must maintain either that constitution is not identity orthat modal predicates are Abelardian. The latter option may be thefruitful one, since for one thing it seems to have applications thatgo beyond the issue of constitution.

2.5 The Ship of Theseus Paradox

Imagine a wooden ship restored by replacing all its planks and beams(and other parts) by new ones. Plutarch reports that such a shipwas

… a model for the philosophers with respect to the disputedarguments … some of them saying it remained the same, some ofthem saying it did not remain the same. (Plutarch,Life ofTheseus, 23; translated by Michael Rea in Rea 1995, p. 531).

Hobbes added the catch that the old parts are reassembled to createanother ship exactly like the original. Both the restored ship and thereassembled one appear to qualify equally to be the original. In theone case, the original is “remodeled”, in the other, it isreassembled. Yet the two resulting ships are clearly not the sameship.

Some have proposed that in a case like this our ordinary“criteria of identity” fail us. The process of dismantlingand reassembling usually preserves identity, as does the process ofpart replacement (otherwise no soldier could be issued just one rifleand body shops would function as manufacturers). But in this case thetwo processes produce conflicting results: We get two ships, one ofwhich is the same ship as the original, by one set of criteria, andthe other is the original ship, by another set of criteria. There is asimilar conflict of criteria in the case of personal identity: Brainduplication scenarios (Wiggins 1967, Parfit 1984) suggest that it islogically possible for one person to split into two competitors, eachwith equal claim to be the original person. We take it for grantedthat brain duplication will preserve the psychological propertiesnormally relevant to reidentifying persons and we also take it forgranted that the original brain continues to embody these propertieseven after it is duplicated. In this sense there is a conflict ofcriteria. Such a case of “fission” gives us two distinctembodiments of these properties.

Perhaps we should conclude that identity is not what matters. Instead,what matters is someother relation, but one that accounts asreadily as identity for such facts as that the owner of the originalship would be entitled to both the restored version and thereassembled one. For the case of personal identity, Parfit (1984)develops such a response in detail. A related reaction would be toclaim that if both competitors have equal claim to be the original,then neitheris the original. If, however, one competitor isinferior, then the other wins the day and counts as the original. Itseems that on this view certain contingencies can establish or falsifyidentity claims. That conflicts with NI. Suppose that \(w\) is apossible world in which no ship is assembled from the discarded partsof the remodeled ship. In this world, then, the remodeled ship is theoriginal. By NI, the restored ship and the original are identical inthe actual world, contrary to the claim of the “bestcandidate” doctrine (which says that neither the remodeled northe reassembled ship is the original). There are, however, moresophisticated “best candidate” theories that are notvulnerable to this objection (Nozick 1982).

Some are convinced that the remodeled ship has the best claim to bethe original, since it exhibits a greater degree of spatio-temporalcontinuity with the original (Wiggins 1967). But it is unclear why theintuition that identity is preserved by spatio-temporal continuityshould take precedence over the intuition that identity is preservedin the process of dismantlement and reassembly. Furthermore, certainversions of the ship of Theseus problem do not involve the featurethat one of the ships competing to be the original possesses a greaterdegree of spatio-temporal continuity with the original than does theother (see below). Others are equally convinced that identity isnot preserved by total part replacement. This view is oftensuggested blindly, as a stab in the dark, but there is in fact aninteresting argument in its favor. Kripke (1980) argues that a tablemade out of a particular hunk of woodcould not have beenmade out of a (totally) different hunk of wood. His reasoning is this:Suppose that in the actual world a table \(T\) is made out of a hunkof wood \(H\); and suppose that there is a possible world \(w\) inwhich this very table, \(T\), is made out of a different hunk of wood,\(H'\). Then assuming that \(H\) and \(H'\) are completely unrelated(for example, they do not overlap), so that making a table out of theone is not somehow dependent upon making a table out of the other,there is another possible world \(w'\) in which \(T\), as in theactual world, is made out of \(H\), and another table \(T'\), exactlysimilar to \(T\), is made out of \(H'\). Since \(T\) and \(T'\) arenot identical in \(w'\), it follows by ND that the table made out of\(H'\) in \(w\) is not \(T\). Note, however, that the argument assumesthat the table made out of \(H'\) in \(w'\) is the same table as thetable made out of \(H'\) in \(w\).

Kripke’s reasoning can be applied to the present case (Kripkeand others might dispute this claim; see below). Let \(w\) be apossible world just like the actual world in that \(O\), the originalship, is manufactured exactly as it is in the actual world. In \(w\),however, another ship, \(S'\), exactly similar to \(O\), issimultaneously built out of precisely the same parts that \(S\), theremodeled ship, is built out of in the actual world. Since \(S'\) and\(O\) are clearly different ships in \(w\), it follows by ND that\(O\) and \(S\) are not the same ship in the actual world. Note againthat the argument assumes that \(S\) and \(S'\) are the same ship, butit seems quite a stretch to deny that. Nevertheless, some have doneso. Carter (1987) claims (in effect) that \(S\) and \(S'\) are notidentical, but his argument simplyassumes that \(O\) and\(S\) are the same ship. Alternatively, one might view the (Kripkean)argument as showing only that while \(S\) is the same ship as \(O\) inthe actual world, \(S\) (that is, \(S')\) is not the same ship as\(O\) in \(w\). But this is not an option for one who adheres to thestandard account and hence adheres to ND. In defending this view,however, Gallois (1986, 1988) suggests a weakened notion of rigiddesignation and a corresponding weakened formulation of ND. (SeeCarter 1987 for criticism of Gallois’ proposals. See alsoChandler 1975 for a precursor of Gallois’ argument.)

If we grant that \(O\) and \(S\) cannot be the same ship, we seem tohave a solution to the ship of Theseus paradox. By the Kripkeanargument, only the reassembled ship has any claim to being theoriginal ship, \(O\). But this success is short-lived. For we are leftwith the following additional paradox: Suppose that \(S\) eventuatesfrom \(O\) by replacing one part of \(O\) one day at a time. Thereseems to be widespread agreement that replacing just one part of athing by a new exactly similar part preserves the identity of thething. If so, then, by the transitivity of identity, \(O\) and \(S\)must be the same ship. It follows that either the Kripkean argument isincorrect, or replacement of even a single part (or small portion)does not preserve identity (a view known as “mereologicalessentialism”; Chisholm 1973).

As indicated, Kripke denies that his argument (for the necessity oforigin) applies to the case of change over time: “The questionwhether the table could havechanged into ice is irrelevanthere” (Kripke 1972, p. 351). So the question whether \(O\) couldchange into \(S\) is supposedly “irrelevant”. But Kripkedoes not give a reason for this claim, and if cases of trans-temporalidentity and trans-world identity differ markedly in relevantrespects—respects relevant to Kripke’s argument for thenecessity of origin—it is not obvious what they are. (But seeForbes 1985, and Lewis 1986, for discussion.) The argument above wassimply that \(O\) and \(S\) cannot be the same ship since there is apossible world in which they differ. If this argument is incorrect itis no doubt because there are conclusive reasons showing that \(S\)and \(S'\) differ. Even so, such reasons are clearly not“irrelevant”. One may suspect that, if applied to thetrans-temporal case, Kripke’s reasoning will yield an argumentfor mereological essentialism. Indeed, a trans-world counterpart ofsuch an argument has been tried (Chandler 1976, though Chandler viewshis argument somewhat differently). In its effect, this argument doesnot differ essentially from the “paradox” sketched in theprevious paragraph (which may well be viewed as an argument formereological essentialism). Subsequent commentators, e.g., Salmon(1979) and Chandler (1975, 1976), do not seem to take Kripke’sadmonition of irrelevance seriously.

In any case, there \(is\) a close connection between the two issues(the ship of Theseus problem and the question of the necessity oforigin). This can be seen (though it may already be clear) byconsidering a modified version of the ship of Theseus problem. Supposethat when \(O\) is built, another ship \(O'\), exactly like \(O\), isalso built. Suppose that \(O'\) never sets sail, but instead is usedas a kind of graphic repair manual and parts repository for \(O\).Over time, planks are removed from \(O'\) and used to replacecorresponding planks of \(O\). The result is a ship \(S\) made whollyof planks from \(O'\) and standing (in the end), we may suppose, inexactly the place \(O'\) has always stood. Now do \(O\) and \(O'\)have equal claim to be \(S\)? And can we then declare that neither is\(S\)? Not according to the Kripkean line of thought. It looks for allthe world as though the process of “remodeling” \(O\) isreally just an elaborate means of dismantling and reassembling \(O'\).And if \(O'\) and \(S\) are the same ship, then since \(O\) and \(O'\)are distinct, \(O\) and \(S\) cannot be the same ship.

This argument is vulnerable to the following two important criticisms:First, it conflicts with the common sense principle that (1) thematerial of an object can be totally replenished or replaced withoutaffecting its identity (Salmon 1979); and secondly, as mentioned, itconflicts with the additional common sense principle that (2)replacement by a single part or small portion preserves identity.These objections may seem to provide sufficient grounds for rejectingthe Kripkean argument and perhaps restricting the application ofKripke’s original argument for the necessity of origin (Noonan1983). There is, however, a rather striking problem with (2), and itis unclear whether the conflict between (1) and the Kripkean argumentshould be resolved in favor of the former.

The problem with (2) is this. Pick a simple sort of objects, say,shoes, or better, sandals. Suppose \(A\) and \(B\) are two exactlysimilar sandals, one of which \((A)\) is brand new and the other\((B)\) is worn out. Each consists of a top strap and a sole, nothingmore. If \(B\)’s worn strap is replaced by \(A\)’s newone, (2) dictates that the resulting sandal is \(B\)“refurbished”. In fact, if the parts of \(A\) and \(B\)are simply exchanged, (2) dictates that the sandal with the new parts,\(A'\), is \(B\) and the sandal with the old parts, \(B'\), is \(A\).It follows by ND that \(A\) and \(A'\) and \(B\) and \(B'\) aredistinct. This is surely the wrong result. The intuition that \(A\)and \(A'\) are the same sandal is very strong; and the process ofexchanging the parts of \(A\) and \(B\) seems to amount to nothingmore than the dismantling and reassembling of each. This example is nodifferent in principle than the more elaborate trans-world casesdiscussed by Chisholm (1967), Chandler (1976), Salmon (1979), or Gupta(1980). (One who claims that \(A\) and \(A'\) differ in that \(A'\)comes into existence after \(A\), does not have much to go on. \(A\)cannot be supposed to persist after \(A'\) comes into existence. We donot end up with twonew sandals and one old one. Why thencouldn’t it be \(A\) itself that reappears at the latertime?)

2.6 Church’s Paradox

The following paradox—perhaps the ultimate paradox ofidentity—derives from an argument of Church (1982). SupposePierre thinks that London and Londres are different cities, but ofcourse doesn’t think that London is different from London, orthat Londres is different from Londres. Assuming that proper nameslack Fregean senses, we can apply LL to get the result that London andLondresare distinct. We have here an argument that, giventhe standard account of identity, merelythinking that \(x\)and \(y\) are distinct is enough to make them so. There are, ofcourse, a number of ways around this conclusion without abandoning thestandard account of identity. Church himself saw the argument (hisversion of it) as demonstrating the inadequacy of Russellianintensional logic—in which variables and constants operate asthey do in extensional logic, i.e., unequipped with senses. (Foranother reaction, see Salmon 1986.) But there are strong argumentsagainst the view that names (or variables) have senses (Kripke 1980).In light of these arguments, Church’s argument may be viewed asposing yet another paradox of identity.

The general form of Church’s argument has been exploited byothers to reach further puzzling conclusions. For example, it has beenused to show that there can be no such thing as vague or“indeterminate” identity (Evans 1978; and for discussion,Parsons 2000). For \(x\) is not vaguely identical to \(x\); hence, if\(x\) is assumed to be vaguely identical to \(y\), then by LL, \(x\)and \(y\) are (absolutely) distinct. As it stands, Evans’argument shows at best that vaguely identical objects must beabsolutely distinct, not that there is no such thing as vagueidentity. But some have tried to amend the argument to getEvans’ conclusion (Parsons 2000; and see the entry onvagueness). In any case, it is useful to see the connection betweenEvans’ argument and Church’s. If, for example,“vaguely identical” is taken to mean “thought to beidentical”, then the two arguments collapse into one another.Church’s line of argument would seem to lead ultimately to theextreme antirealist position that any perceived difference amongobjects is a real difference. If one resolves not to attempt to escapethe clutches of LL by some clever dodge—by disallowingstraightforward quantifying-in, for example, as with the doctrine ofAbelardian predicates—one comes quickly to the absurd conclusionthat no statement of the form \(x = y\), where the terms aredifferent, or are just different tokens of the same type, can be true.Yet it might just be that the fault lies not in ourselves, but inLL.

2.7 The Theological Paradox

The Christian doctrine of Trinity is sometimes construed as a paradoxthat involves incongruent identity statements. For example, theAthanasian Creed puts it like this: “[…] we worship oneGod in Trinity. […] the Father is God; the Son is God; and theHoly Spirit is God. And yet they are not three Gods; but oneGod”. One may conceptually unbundle these statements intothe following set of claims:

1. The Father is a God.
2. The Son is a God.
3. The Holy Spirit is a God.
4. The Father is not identical with the Son.
5. The Father is not identical with the Holy Spirit.
6. The Son is not identical with the Holy Spirit.
7. There is at most one God, i.e., if \(x\) is a God and \(y\) is aGod, then \(x\) is identical with \(y\).

It is easy to see that this set is logically inconsistent andconsequently one may claim that the doctrine itself is conceptuallyincongruent.

3. Relative Identity

The fundamental claim of relative identity—the claim the variousversions of the idea have in common—is that, as it seems in thepassenger/person case, it can and does happen that \(x\) and \(y\) arethe same \(F\) and (yet) \(x\) and \(y\) are not the same \(G\). Nowit is usually supposed that if \(x\) and \(y\) are the same \(F\)(\(G\) etc.), then that implies that \(x\) and \(y\) are \(F\)s\((G\)s, etc.) If so, then the above schema is trivially satisfied bythe 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 thephrase “\(x\) and \(y\) are different \(G\)s” to mean“\(x\) and \(y\) are \(G\)s and \(x\) and \(y\) are not the same\(G\)”. Then the nontrivial core claim about relative identityis that the following may well be true:

(RI)

\(x\) and \(y\) are the same \(F\) but \(x\) and \(y\) aredifferent \(G\)s.

RI is a very interesting thesis. It seems to yield dramatically simplesolutions to (at least some of) the puzzles about identity. We appearto be in a position to assert that young Oscar and old Oscar are thesame dog but nonetheless distinct “temporary” objects;that Oscar and Oscar⁻ are the same dog but differentdog parts; that the same piece of clay can be now (identical to) onestatue and now another; that London and Londres are the same city butdifferent “objects of thought”, and so forth. Doubtsdevelop quickly, however. Either thesame dog relationsatisfies LL or it does not. If it does not, it is unclear why itshould be taken to be a relation ofidentity. But if itsatisfies LL, then it follows, given that Oscar andOscar⁻ are different dog parts, thatOscar⁻ is not the same dog part asOscar⁻. Furthermore, assuming that thesame dogpart relationis reflexive, it follows from theassumption that Oscar⁻ and Oscar⁻are the same dog (and that LL is in force), that Oscar andOscar⁻ are indeed the same dog part, which in factthey 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 thefollowing 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 relationx and y are the same F figuring in (1)satisfies LL.

For suppose that the relationx and y are the same G,figuring in (1), is reflexive and that \(x\) is a \(G\). Then \(x\) isthe same \(G\) as \(x\). But according to (1), \(x\) and \(y\) are notthe same \(G\)s; 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 identityrelations are equivalence relations, they do not in general satisfyLL. However, according to at least one analysis of thepassenger/person case (and others), thesame person relationsatisfies LL but thesame passenger relation is notstraightforwardly an equivalence relation (Gupta 1980). It should beclear though that this view is incompatible with the principle of theidentity of indiscernibles: If \(x\) and \(y\) are differentpassengers, there must be, by the latter principle, some property\(x\) possesses that \(y\) does not. Hence if thesame personrelation satisfies LL, it follows that \(x\) and \(y\) arenot the same person. For the remainder we will assume thatidentity relations are equivalence relations. Given this assumption(and assuming that the underlying propositional logic isclassical—cf. Parsons 2000), RI and LL are incompatible in thesense that within the framework of a single fixed language for whichLL is defined, RI and LL are incompatible.

Yet the advocate of relative identity cannot simply reject any form ofLL. 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 motivatingrestricted forms of LL that are nonetheless strong enough tobear the burden of identity claims. There has been little systematicwork done in this direction, crucial though it is to the relativeidentity project. (See Deutsch 1997 for discussion of this issue.)There are, however, equivalence relations that do satisfy restrictedforms of LL. These are sometimes called “congruencerelations” and they turn up frequently in mathematics. Forexample, say that integers \(n\) and \(m\) are congruent if theirdifference \(n - m\) is a multiple of 3. This relation preservesmultiplication and addition, but not every property. The numbers 2 and11 are thus congruent but 2 is even and 11 is not. There are alsonon-mathematical congruencies. For example, the relationx and yare traveling at the same speed preserves certain properties andnot others. If objects \(x\) and \(y\) are traveling at the same speedand \(x\) is traveling faster than \(z\), the same is true of \(y\).Such similarity relations satisfy restricted forms of LL. In fact, anyequivalence relation satisfies a certain minimal form of LL (seebelow).

There are strong and weak versions of RI. The weak version says thatRI has some (in fact, many) true instances but also that there arepredicates \(F\) such that if \(x\) and \(y\) are the same \(F\),then, for any equivalence relation, \(E\), whatsoever (whether or notan identity relation), \(E(x,y)\). This last condition implies thatthe relationx and y are the same F satisfies LL. Therelation \(P\) 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 anequivalence relation (namely, \(P(x,y))\) that fails to hold of \(x\)and \(y\). If we add that in this instance “\(x\) and \(y\) arethe 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 athing as relative identity and such a thing as absolute identity aswell. The strong version, by contrast, says that there are (many) trueinstances of RI but there is no such thing as absolute identity. It isdifficult 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 demonstrablycoherent. We return to this matter in §5.

The puzzles about identity outlined in §2 (and there are manyothers, as well as many variants of these) put considerable pressureon the standard account. A theory of identity that allows forinstances of RI is an attractive alternative (see below §4). Butthere is a certain kind of example of RI, frequently discussed in theliterature, 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 relationalexpression “passenger in (on) …”. A passenger issomeone who is a passenger in some vehicle (on some flight, etc.).Similarly, a father is a man who fathers someone or who is the fatherof someone. This way of defining a kind of things from a relationbetween things is perfectly legitimate and altogether open-ended.Given any relation \(R\), we can define “an \(R\)” toapply to anything \(x\) that stands in \(R\) to something \(y\). Forexample, we can define a “schmapple” to be an apple in abarrel. All this is fine. But we can’tinfer from sucha definition that the same apple might be two different schmapples.From the fact that someone is the father of two different children, wedon’t judge that he is two different fathers. The fact thatairlines choose to count passengers as they do, rather than trackpersons, is their business, not logic’s.

However, when \(R\) is an equivalence relation, we are entitled tosuch an inference. Consider the notorious case of “surmen”(Geach 1967). A pair of men are the “same surman” if theyhave the same surname; and a surman is a man who bears this relationto someone. So now it appears that two different men can be the samesurman, since two different men can have the same surname. As Geach(1967) insists (also Geach 1973), surmen aredefined to bemen, so they are not merely classes of men. Hence we seem to have aninstance of RI, and obviously any similarity relation (e.g., \(x\) and\(y\) have the same shape) will give rise to a similar case. Yet suchinstances of RI are not very interesting. It is granted all aroundthat when“\(F\)” is adjectival,different \(G\)s may be the same \(F\). Different men may have thesame surname, different objects the same color, etc. Turning anadjectival similarity relation into a substantival one having the formof an identity statement yields an identity statement in nameonly.

A word about the point of view of those who subscribe to the weakversion of RI. The view (call it the “weak view”) is thatordinary identity relations concerning (largely) the world ofcontingency and change are equivalence relations answering torestricted forms of LL. The exact nature of the restriction depends onthe equivalence relation itself, though there is an element ofgenerality. The kinds of properties preserved by thesame dogrelation are intuitively the samekinds of properties as arepreserved by thesame cat relation. From a logical point ofview the best that can be said is that any identity relation, like anyequivalence relation, preserves a certain minimal set of properties.For suppose \(E\) is some equivalence relation. Let \(S\) be the setcontaining all formulas of the form \(E(x,y)\), and closed under theformation 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 instanceof RI implies that \(x\) and \(y\) are absolutely distinct.

4. The Paradoxes Reconsidered

Let us look back at the paradoxes of identity outlined in §2 fromthe perspective of the weak view regarding relative identity. Thatview 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 relativelydifferent objects, then \(x\) and \(y\) are absolutely distinct, andhence 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. Absolutelydistinct apples, for example, are distinct logical objects.

The following is the barest sketch of relativist solutions to theparadoxes of identity discussed in §2. No attempt is made tofully justify any proposed solution, though a modicum of justificationemerges in the course of §6. It should be kept in mind that someof the strength of the relativist solutions derives from theweaknesses of the absolutist alternatives, some of which are discussedin §2.

4.1 The Paradox of Change

Young Oscar and old Oscar are the same dog but absolutely differentthings, i.e., different logical objects. The material conditionsrendering young Oscar and old Oscar the same dog (and the sameDalmatian) are precisely the same as the material conditions underwhich young Oscar and old Oscar would qualify as temporal parts of thesame dog. The only difference islogical. The identityrelation between young Oscar and old Oscar can be formalized in anextensional logic (Deutsch 1997), but a theory of temporal partsrequires a modal/temporal apparatus. Young Oscar is wholly presentduring his youth and possesses the simple, non-relational, property ofnot having a gray muzzle.

4.2 Chrysippus’ Paradox

Oscar and Oscar⁻ both survive Oscar’s loss of atail. At both \(t\) and \(t'\) Oscar and Oscar⁻ arethe same dog, but at \(t\), Oscar and Oscar⁻ aredistinct logical objects. This implies (by ND) that Oscar andOscar⁻ are distinct logical objects even at \(t'\)Hence, we must allow that distinct logical objects may occupy the samespace at the same time. This is not a problem, however. For althoughOscar and Oscar⁻ are distinct logical objects at\(t'\), they are physically coincident.

4.3 The Paradox of 101 Dalmatians

The relativist denies that dogs are “maximal”. It is nottrue 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 ahair are dogs. In fact, many (though of course not all) identitypreserving changes Oscar might undergo correspond directly to properparts of (an unchanged) Oscar. But there is no problem about barkingin unison, and no problem about acting independently. All 101 are thesame dog, despite their differences, just as young Oscar and old Oscarare the same dog. The relativist denies that the dogs are many ratherthan deny that the many are dogs (Lewis 1993).

4.4 The Paradox of Constitution

Constitution is identity,absolute identity. The relationbetween the piece of clay \(c\) and the statue \(s_1\) on day 1 is oneof absolute identity. So we have that \(c = s_1\) on day 1, and forthe same reason, \(c = s_2\) on day 2. Furthermore, since \(s_1\) and\(s_2\) are different statues, it follows (on the weak view) that\(s_1\ne s_2\). In addition, the piece of clay \(c\) constituting\(s_1\) on day 1 is (relatively) the same piece of clay as the pieceof clay constituting \(s_2\) on day 2. (The identity is relativebecause we have distinct objects—the two statues—that arethe same piece of clay.) It follows thatno name of the piece ofclay 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 day2. For on day 1, a name of the piece of clay \(c\) would denote\(s_1\) and on day 2, it would denote \(s_2\), and \(s_1\) and \(s_2\)are absolutely distinct. Nevertheless, a name of the piece of clay mayberelatively rigid: it may denote at each time thesamepiece of clay. Although no name of the piece of clay \(c\) isabsolutely rigid, that does not prevent the introduction of a name of\(c\) that denotes \(c\) at any time (or possible world). (Kraut 1980discusses a related notion of relative rigidity.)

There is, however, a certain ambiguity in the notion of a name of thepiece of clay, inasmuch as the piece of clay may be any number ofabsolutely distinct objects. The notion of relative rigiditypresupposes that a name for the piece of clay refers, with respect tosome parameter \(p\), to whatever object counts as the piece of clayrelative to that parameter. This may be sufficient in the case of thepiece of clay, but in other cases it is not. With respect to a fixedparameter \(p\) there may be no unique object to serve as the referentof the name. For example, if any number of dog parts count, at a fixedtime, as the same dog, then which of these objects serves as thereferent of “Oscar”? We shall leave this question open forthe time being but suggest that it may be worthwhile to view namessuch as “Oscar” asinstantial terms—termsintroduced into discourse by means of existential instantiation. Thename “Oscar” might be taken as denoting a representativemember of the equivalence class of distinct objects qualifying as thesame dog as Oscar. It would follow, then, that most ordinary names areinstantial terms. (An alternative is that of Geach 1980, who draws adistinction between aname of and aname for anobject; see Noonan 1997 for discussion of Geach’sdistinction.)

4.5 The Ship of Theseus Paradox

In this case, the relativist, as so far understood, may seem to enjoyno advantage over the absolutist. The problem is not clearly one ofreconciling LL with ordinary judgments of identity, and the advantageafforded by RI does not seem applicable. Griffin (1977), for example,relying on RI, claims that the original and remodeled ship are thesame ship but not the same collection of planks, whereas thereassembled ship is the same collection of planks as the original butnot the same ship. This simply doesn’t resolve the problem. Theproblem is that the reassembled and remodeled ships have,primafacie, equal claim to be the original and so the bald claims thatthe reassembled ship is not—and the remodeled ship is—theoriginal are unsupported. The problem is that of reconciling theintuition that certain small changes (replacement of a single part orsmall portion) preserve identity, with the problem illustrated by thesandals example of §2.5. It turns out, nevertheless, that theproblem \(is\) one of dealing with the excesses of LL. To resolve theproblem, we need an additional level of relativity. To motivate thisdevelopment, consider the following abstract counterpart of thesandals example:

On the left there is an object \(P\) composed of three parts, \(P_1,P_2\), and \(P_3\). On the right is an exactly similar butnon-identical object, \(Q\), composed of exactly similar parts, \(Q_1,Q_2\), and \(Q_3\), in exactly the same arrangement. For the sake ofillustration, we adopt the rule that only replacement of (at most) asingle part by an exactly similar part preserves identity.Suppose we now interchange the parts of \(P\) and \(Q\). We begin byreplacing \(P_1\) by \(Q_1\) in \(P\) and replacing \(Q_1\) by \(P_1\)in \(Q\), to obtain objects \(P^1\) and \(Q^1\). So \(P^1\) iscomposed of parts \(Q_1,\) \(P_2,\) and \(P_3\), and Q\(^1\) iscomposed of parts \(P_1,\) \(Q_2,\) and \(Q_3\). We then replace\(P_2\) in \(P^1\) by \(Q_2\), to obtain \(P^2\), and so on. Given oursample criterion of identity, and assuming the transitivity ofidentity, \(P\) and \(P^3\) are counted the same, as are \(Q\) and\(Q^3\). But this appears to be entirely the wrong result.Intuitively, \(P\) and \(Q^3\) are the same, as are \(Q\) and \(P^3\).For \(P\) and \(Q^3\) are composed of exactly the same parts puttogether in exactly the same way, and similarly for \(Q\) and \(P^3\).Furthermore, \(Q_3 (P_3)\) can be viewed as simply the result oftaking \(P (Q)\) apart and puttingit back together in aslightly different location. And this last difference can beeliminated by switching the locations of \(P^3\) and \(Q^3\) as a laststep in the process.

Suppose, however, that we replace our criterion of identity by thefollowing more complicated rule: \(x\) and \(y\) are the samerelative to z, if both \(x\) and \(y\) differ from \(z\) atmost by a single part. (This relation is transitive, and is in fact anequivalence relation.) For example,relative to \(P\), \(P,P^1, Q^2\), and \(Q^3\) are the same, but \(Q, Q^1, P^2\) and \(P^3\),are not. Of course, replacement by a single part is an artificialcriterion of identity. In actual cases, it will be a matter of thedegree or kind of deviationfrom the original (represented bythe third parameter, \(z)\). The basic idea is that identity throughchange is not a matter of identity through successive, accumulatedchanges—that notion conflicts with both intuition (e.g., thesandals example) and the Kripkean argument: Through successive changesobjects can evolve intoother objects. The three-placerelation of identity does not satisfy LL and is consistent with theoutlook of the relativist. Gupta (1980) develops a somewhat similaridea in detail. Williamson (1990) suggests a rather differentapproach, but one that, like the above, treats identity through changeas an equivalence relation that does not satisfy LL.

4.6 Church’s Paradox

Church’s argument implies that if Pierre’s doxasticposition is as described (in §2.6), then London and Londres aredistinct objects. Assuming the standard account of identity, theresult is that either Pierre’s doxastic positioncannotbe as described or else London and Londres are differentcities (or else we must punt). Since London and Londres arenot different cities, the standard account entails that Pierre’sdoxastic position cannot be as described (or else we must punt). Thiswas Church’s own position as regards certain puzzles aboutsynonymy, such as Mates’ puzzle (Mates 1952). Church held thatone who believes that lawyers are lawyers, must indeed believe thatlawyers are attorneys, despite any refusal to assent to (or desire todissent from) “Lawyers are attorneys” (Church 1954).Kripke later argued (Kripke 1979) that assent and failure to assentmust be taken at face value (at least in the case of Pierre) andPierre’s doxastic position is as described. Kripke chose topunt—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 Londresare nonetheless the same city. Whether this resolution ofChurch’s paradox can be exploited to yield solutions toFrege’s puzzle (Salmon 1986) or Kripke’s puzzle (1979)remains to be seen. Crimmins (1998) has suggested that the analysis ofpropositional attitudes requires a notion of “semanticpretense”. In reporting Pierre’s doxastic position weengage in a pretense to the effect that London and Londres aredifferent 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 (Kripke1979). The relativist proposes just such a reconciliation but suggeststhat the pretense can be dropped.

4.7 The Theological Paradox

The solution provided by RI is straightforward: we need to state thedoctrine not in terms of absolute identity, but in terms of relativeidentity. So instead of one predicate of absolute identity thetrinitarian creed may need two predicates of relative identity, e.g.,being the same person and being the same being (cf. van Inwagen 1988,2003), or, as suggested by (Anscombe and Geach 1961), being the sameperson and being the same God. Then we may reformulate the creed asfollows:

1. The Father is a God.
2. The Son is a God.
3. The Holy Spirit is a God.
4. The Father is not the same person as the Son.
5. The Father is not the same person as the Holy Spirit.
6. The Son is not the same person as the Holy Spirit.
7. There is at most one God, i.e., if \(x\) is a God and \(y\) is aGod, then \(x\) is the same being as \(y\).

Van Inwagen produced a model (in the sense of formal semantics) whereall these statements are satisfied, so under this interpretation thedoctrine of Trinity comes out consistent. Incidentally let me notethat he also gave an analogous interpretation for the doctrine ofIncarnation (see van Inwagen 1994). For an extended discussion of thecoherence of this solution see the section on relative identitytheories in the entry onTrinity. Note that Branson (2019) provides a taxonomy of all possiblesolutions to the paradox, including the ones that resort to relativeidentity.

5. Absolute Identity

The philosopher P.T. Geach first broached the subject of relativeidentity and introduced the phrase “relative identity”.Over the years, Geach suggested specific instances of RI (a variant ofthe case of Oscar and his tail is due to Geach 1980) and in this wayhe contributed to the development of the weak view concerning relativeidentity, i.e., the view that while ordinary identity relations areoften relative, some are not. But Geach maintains that absoluteidentity does not exist. What is his argument?

That is hard to say. Geach sets up two strawman candidates forabsolute identity, one at the beginning of his discussion and one atthe end, and he easily disposes of both. In between he develops aninteresting and influential argument to the effect that identity,even as formalized in the system FOL⁼, is relativeidentity. However, Geach takes himself to have shown, by thisargument, that absolute identity does not exist. At the end of hisinitial presentation of the argument in his 1967 paper, Geachremarks:

We thought we had a criterion for a predicable’s expressingstrict identity [i.e., as Geach says, “strict, absolute,unqualified identity”]; but the thing has come apart in ourhands; and no alternative rigorous criterion that could replace thisone has thus far been suggested. (Geach 1967, p. 6 [1972, p. 241])

It turns out, as we’ll see, that all that comes apart is thefalse notion that in FOL⁼ the identity symboldefines the relation \(I(A,x,y)\). Let us examineGeach’s line of reasoning in detail, focusing on thepresentation in his 1967 article, thelocus classicus of thenotion of relative identity.

Geach begins by urging that a plain identity statement “\(x\)and \(y\) are the same” is in need of a completing predicate:“\(x\) and \(y\) are the same \(F\)”. Frege had arguedthat statements of number such as “this is one” require acompleting predicate: “this is one \(F\)”, and so it is,Geach claims, with identity statements. This is a natural view for onewho subscribes to RI. The latter cannot even be stated without thecompleting predicates. Nevertheless, both the claim itself and theanalogy with Frege have been questioned. Some argue that the analogywith Frege is incorrect. For example, Carrara and Sacchi (2007)maintain that Frege’s position that any ascription of a numberto something is always relative to a concept boils down to the claimthat concepts are essential in cardinality statements because withoutthem there is no specification of what is to be counted. Once this issettled, Frege would count by means of absolute identity. Othersargued that while the analogy is correct, both Frege and Geach arewrong (Perry 1978 and Alston and Bennett 1984). These matters do notbear directly on the question of the coherence and truth of RI or thequestion of absolute identity. One who adopts the weak view would notwant to follow Geach on this score. And one could maintain the“completing thesis” without being committed to RI.Furthermore, the completing thesis occupies a puzzling role inGeach’s dialectic. Immediately following his statement of thethesis, Geach formalizes FOL⁼ on the basis of the singleformula:

(R)

\(\phi(a) \leftrightarrow \exists x(\phi(x) \wedge x = a)\)

(The “W” is for Hao Wang, who first suggested it. Thereader is invited to prove Ref and LL from W.) But we hear nocomplaint about the syntax of W despite its involving a seeminglyunrelativized identity symbol. It turns out, however, that Geachapparently thinks of the completing predicate as being given by thewhole descriptive apparatus of \(L\) or a fragment thereof.

Geach now observes

… if we consider a moment, we see that an I-predicable in agiven theoryTneed not express strict, absolute,unqualified identity; it need mean no more than that two objects areindiscernible by the predicables that form the descriptive resourcesof the theory—theideology of the theory ….(1967, p. 5 [1972, p. 240])

Here an “I-predicable” is a binary relation symbol“=” satisfying (W). Geach’s focus at this point ison the need to relativize an I-predicable to a theory \(T\). Geachthen immediately saddles the friend of absolute identity with the viewthat for “real identity” we need not bring in the ideologyof a definite theory. This is Geach’s first strawman. Whenlogicians, in discussing FOL⁼, speak of “realidentity”—and they often do (see Enderton 2000 or Silver1994, for example)—they do not mean a relation ofuniversalidentity, since the universal set does not exist. Nor do theyintend, in formulating LL, to use “true of” in acompletely unrestrained way which gives rise to semantic paradox. Itis no argument against those who wish to distinguish mereindiscernibility from real identity to say that they “will soonfall into contradictions”, e.g., Grelling’s orRussell’s. The relation \(I(A,x,y)\) is sufficientlyrelativized. (It is relativized to aset A.)

We come next to the main point:

Objects that are indiscernible when we are confined to the ideology of\(T\) may perfectly well be discernible in the ideology of a theory\(T^1\) of which \(T\) is a fragment. (Geach 1967, p. 5 [1972, p.240])

The warrant for this claim can only be the language relativity ofidentity when treated as a non-logical notion (see §1). That thisis what Geach has in mind is clear from some approving remarks hemakes in his 1973 article about Quine’s (1970) proposal to treatidentity as a non-logical notion. But how does it follow that absoluteidentity does not exist? Geach seems to think that the defender ofabsolute identity will look to Ref and LL (or W)—and notbeyond—for a full account of “strict, absolute,unqualified” identity. That is not so. The fact that theseformulas in themselves define only indiscernibility relations is alogical commonplace. So this is Geach’s second strawman.

Is Geach’s argument at least an argument that identity isrelative? Does language relativity support the conclusion that RI istrue even of identity as formalized in FOL⁼? The generalidea appears to be that language relativity suggests that we takeidentityto be indiscernibility, and conclude that objectsidentical relative to one ideology \(F\) may be different relative toanother ideology \(G\), and that this confirms RI. Notice first of allthat this argument relies on the identity of indiscernibles: thatindiscernibility implies identity. This principle is not valid inFOL⁼ even when the latter is treated as a proper theory.Language relativity does not imply that the distinctness of distinctobjects cannot go unnoticed.

Secondly, the interesting cases of RI do not involve a shift from animpoverished point of view to an improved one—whether this isseen in epistemic terms (which Geach disputes—Geach 1973) or inpurely logical terms. We do not affirm that old Oscar and young Oscar,for example, are the same dog on the grounds that there is an ideologywith respect to which old Oscar and young Oscar are indistinguishable.Such an ideology would be incapable of describing any change in Oscar.It is true that thesame dog relation determines a set ofpredicates that do not discriminate between the members of certainpairs of dogs—the dogs in the photographs mentioned earlier, forexample. And it is true that these predicates determine a sublanguagein which thesame dog relation is a congruence, i.e., nopredicate of the sublanguage distinguishes \(x\) from \(y\), if \(x\)and \(y\) are the same dog. But the verysense of suchstatements as that old Oscar and young Oscar are the same dog requiresa language in which a change in Oscar is expressible. We are talking,after all, aboutold Oscar andyoung Oscar. If wetake seriously the idea that change involves the application ofincompatible predicates, then the sublanguage cannot express thecontrast between old Oscar and young Oscar.

Third, the phenomenon of language relativity (in the technical sensediscussed in §1) has led many philosophers, including Geach, tothe view that ideology creates ontology. There is no antecedentlygiven domain of objects, already individuated, and waiting to bedescribed. Instead, theories carve up the world in various ways,rendering some things noticeably distinct and others indiscernible,depending on a theory’s descriptive resources. The very notionofobject is theory-bound (Kraut 1980). This sort ofanti-realism may seem to go hand in hand with relative identity. Modeltheory, however, is realist to its core and language relativity is amodel-theoretic phenomenon. It is a matter of definability (in astructure). Referring back to §1, in order to make sense oflanguage relativity we have to start with a pair ofdistinctobjects, \(a\) and \(b\) (distinct from the standpoint of themetalanguage), and hence a pair of objects we assume are alreadyindividuated. These objects, however, are indistinguishable in \(M\),since no formula of \(L'\) defines a subset of \(M\) containing theone object and not the other. When we move to \(M'\), we find thatthere is a formula of the enriched language that defines such a subsetin \(M'\). Thus, language relativity is not really any sort ofrelativity ofidentity at all. We must assume that theobjects \(a\) and \(b\) are distinct in order todescribe thephenomenon. If we are living in \(M\), and suspect that Martiansliving in \(M'\) can distinguish \(a\) from \(b\), our suspicion isnot merely to the effect that Martians carve things up differentlythan we do. Our own model theory tells us that there is more to itthan that. Our suspicion must be to the effect that \(a\) and \(b\)are absolutely distinct. If we are blind to the differencebetween \(a\) and \(b\), but the Martians are not, then there must bea difference; and even if we are living in \(M\), we knowthere’s a difference, or at least we can suspect there is, sincemodel theory tells us that such suspicion is well founded.

Let us go back to Geach’s remark that we “neednot” interpret identity absolutely. While this is true, weneed not interpret it as indiscernibility either. There are always thequotient structures (Quine 1963). Instead of taking our“reality” to be \(M\), and our “identity” tobe indiscernibility in \(M\), we can move to the quotient structure,\(\textit{QM}\), whose elements are the equivalence classes, \([x]\),for \(x\) in \(M\). If \(x\) and \(y\) are indiscernible in \(M\),then in \(\textit{QM}\), \([x]\) and [\(y\)] are absolutely identical.We can do this even if we wish to treat FOL⁼ as a propertheory. For example, suppose \(L'\) is a language in which peoplehaving the same income are indiscernible. The domain of \(M\) nowconsists of people. \(\textit{QM}\), however, consists of incomegroups, equivalence classes of people having the same income, andidentity in \(\textit{QM}\) is absolute. Geach objects to suchreinterpretation in terms of the quotient structures on the groundsthat it increases the ontological commitments not only of \(L'\) butof any language of which \(L'\) is a sublanguage.

Let’s focus on \(L'\) first. From a purely model-theoretic pointof view the question is moot. We cannot deny that \(\textit{QM}\) is astructure for \(L'\). Thus, \(L'\) is committed to peoplevisà vis one structure and to income groupsvis àvis the corresponding quotient structure. But let’s pretendthat the structures are “representations of reality”, andso the question now becomes: Which representation is preferable? Isthere then any reason to prefer the ontology of \(M\) to that of\(\textit{QM}\)? \(M\) contains people but no sets of people, whereas\(\textit{QM}\) contains sets of people but no people. ByQuine’s criterion of ontological commitment—that to be isto be the value of a variable—commitment to a set of objectsdoes not carry a commitment to its elements. That is one of the oddconsequences of Quine’s criterion. Unless there is someontological reason to prefer people to sets of people (perhaps becausesets are never to be preferred), the ontologies of \(M\) and\(\textit{QM}\) seem pretty much on a par. Both commit \(L'\) to onekind of thing.

Geach makes the additional claim that the ontological commitments of asublanguage \(L'\) of a language \(L\) are inherited by \(L\) (Geach1973). Suppose then that \(L\) is a language containing expressionsfor several equivalence relations defined on people: say,sameincome,same surname, andsame job. Geachargues that \(L\) need only be committed to the existence of people.Things such as income groups, job groups (equivalence classes ofpeople with the same job), and surmen can all be counted using theequivalencies, without bringing surmen, job groups, and income groupsinto the picture. Consider any sublanguage for which any one of theseequivalence relations is a congruence, i.e., for which\(\textrm{LL}'\) holds. Pick the language, \(L_1\), for example, inwhich people having the same job are indiscernible. More precisely, weassume that \(T_1\) is the pure theory with identity whose ideology isconfined to the language \(L_1\). Let \(M_1\) be a model of \(T_1\).We may imagine the domain of \(M_1\) to consist of people, and we caninterpret indiscernibility in \(M_1\) to be the relationx and yhave the same job. Geach would argue that if \(L_1\) is committedto the elements of \(\textit{QM}_1\)—the job groups—thenso is \(L\). But that is not true. If \(T\) is a theory of the threedistinct equivalence relations formulated in \(L\), the most \(T\) (or\(L)\) would be committed to are the partitions determined by theequivalence relations; and in any case, it would be perfectlyconsistent to insist that, whatever the ontological commitments of\(L_1\), reality, as described by \(L\), consists of people.

The foregoing considerations are rather abstract. To see more clearlywhat is at stake, let us focus on a specific example. Geach (1967)mentions that rational numbers are defined set-theoretically to beequivalence classes of integers determined by a certain equivalencerelation defined on “fractions”, i.e., ordered pairs ofintegers (1/2 is \(\langle 1,2\rangle\), 2/4 is \(\langle2,4\rangle\), etc.). He suggests that we can instead construe ourtheory of rational numbers to be about the fractions themselves,taking the I-predicable of our theory to be the following equivalencerelation, \(E\):

This approach, Geach says, would have “the advantage oflightening a theory’s set-theoretical burdens. (In our presentexample, we need not bring ininfinite sets of ordered pairsof integers into the theory of rationals.)” (Geach 1972, p. 249)

The first thing to notice about this example is thatE cannotbe the I-predicable of such a theory, since \(E\) isdefinedin terms of identity (look at the right side of R). It is“=” that must serve as the I-predicable, and it rendersdistinct ordered pairs of integers discernible. The moral is that notall equivalence relations can be drafted to do the job of identity,even given a limited ideology. There is, indeed, a plausible argumentthatany equivalence relation presupposes identity—notnecessarily in the direct way illustrated by (R), but indirectly,nonetheless (see §6). Moreover, from the standpoint of generalmathematics, once we have (R), we have the (infinite) equivalenceclasses it determines and the partition it induces. These areinescapable. Even from a more limited viewpoint, it seems that once wehave enough set theory to give us ordered pairs of integers and theability to define (R), we get the partition it induces as well.

Geach perceives an ontological advantage in relative identity; but hisargument is unconvincing. Shifting to the quotient structures, asQuine suggested, does not induce a “baroque, Meinongianontology” (Geach 1967, p. 10 [1972, p. 245]). In particular, the“home language” \((L)\) does not inherit the commitment ofthe fragment \((L_1)\), and the ontology of an arbitrary model of thepure theory of identity based on the latter language is at least nomore various than that of the corresponding quotient model. There are,however, a number of ways in which relative identity does succeed inavoiding commitment to certain entities required by its absoluterival. These are discussed in the replies to objections 4 and 5 in thenext section.

6. Objections and Replies

The following constitute a “start-up” set of objectionsand replies concerning relative identity and/or aspects of theforegoing account of relative identity and its rival. Time and spaceconstraints prevent a more extended initial discussion. In addition,there is no presumption that the objections discussed below are themost important or that the initial replies to them are without fault.It is hoped that the present discussion will evolve into a morefull-blown one, involving contributions by the author and readersalike. Should the discussion become lengthy, old or unchallengedobjections 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) whetherthey really are theories of numerical identity, (b) whether they canbe made internally consistent, and (c) whether they are sufficientlymotivated” (Burke 1994, p. 133).

Reply: In reverse order: (c) The issues discussed in§2 and §4 surely provide sufficient motivation. (b) No proofof inconsistency has ever been forthcoming from opponents of relativeidentity, and in fact the weak view is consistent inasmuch as it has amodel in the theory of similarity relations. The arguments outlined inthe second paragraph of §3 are frequently cited as showing thatrelative identity is incoherent; but they show only that RI isincompatible with (unrestricted) LL. (a) See the replies to objections2 and 3 below.

Objection 2: If an identity relation obeys only arestricted form of LL—if it preserves onlysomeproperties and notall—then how do wetellwhich properties serve to individuate a pair of distinct objects?

Reply: Similarity relations satisfy only restrictedforms of LL. How then do wetell which properties arepreserved by thesame shape relation and which are not? It isno objection to the thesis that identity relations in general preservesome properties and not others to demand to know which are which. Atbest the objection points to a problem we must face anyway (for thecase of similarity). In general, a property is preserved by anequivalence relation if it “spreads” in an equivalenceclass determined by the relation: If one member of the class has theproperty, then every member does. Every property spreads in asingleton, as absolute identity demands.

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

Reply: The distinction between identity andsimilaritystatements (or sentences) is usually drawn interms of the distinction between substantival and adjectival commonnouns. If \(F\) is a common noun standing for a kind ofthings, 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 ofpropertiesof things, then “\(x\) and \(y\) are the same \(F\)” is astatement of similarity. (It’s interesting to note that when thenoun is proper, i.e., a proper name, the result is a statement ofsimilarity, not identity—as in “He”s not the sameBill we knew before’.) This distinction rests ultimately on themetaphysical distinction between substance and attribute, object andproperty. While the distinction no doubt presupposes the concept ofindividuation (the bundle theory, for example, presupposes that wehave the means to individuate properties), there is no obvious reasonto suppose that it entails the denial of RI, i.e., the claim that noinstance of RI is true. For a beginner’s review—from anhistorical perspective—of the issues concerning substance andattribute, see O’Connor (1967); and for more recent and advanceddiscussion and bibliography, see the entry onproperties.

Objection 4: Consider the following alleged instanceof RI:

(1)

\(A\) is the same word type as \(B\), but \(A\) and \(B\) aredifferent word tokens.

“If ‘\(A\)’ and ‘\(B\)’ refer to thesame objects throughout (1), the first conjunct of (1) is not anidentity statement, and the counterexample (to the thesis that noinstance of RI is true) fails. If both conjuncts are identitystatements in the required sense, ‘\(A\)’ and‘\(B\)’ must refer to word types in the first conjunct andword tokens in the second, and the counterexample fails” (Perry1970, p. 189).

Reply: First, if “in the required sense”means “satisfies LL”, then the objection buys correctnessonly at the price of begging the question. Advocates of relativeidentity will maintain that the relationA is the same word typeas \(B\) is an identity relation, defined on tokens, that doesnot satisfy LL.

Secondly, even if one insists that in this case intuition dictatesthat if \(A\) and \(B\) refer to tokens in both conjuncts of (1), then“\(A\) is the same word type as \(B\)” expresses only thesimilarity relation:A and B are tokens of the same type,there are other cases where, intuitively, both conjuncts of RI involveidentity relations and yet the relevant terms all refer to the samekind of things; for example,

(2)

\(A\) and \(B\) are the same dog but \(A\) and \(B\) are differentphysical objects,

as said of young Oscar and old Oscar. Here there is no temptation tosuppose that the relationA and B are the same dog isnot an identity relation. One may invoke a theory—atheory of temporal parts, for example—that construes therelation as a certain kind of similarity, but that is theory, notpretheoretical intuition. It is no objection to the relativist’stheory, which holds in part that “\(A\) and \(B\) arethe same dog” expresses a relation of primitive identity, thatthere is an alternative theory according to which it expresses asimilarity relation obtaining between two temporal parts of the sameobject. Furthermore, in the case of (2), \(A\) and \(B\) refer, againintuitively, to the same things in both conjuncts.

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

(3)

\(A\) and \(B\) are the same piece of clay but \(A\) and \(B\) aredifferent statues.

Suppose \(A\) and \(B\) are understood to refer to one sort ofthing—pieces of clay—in the first conjunct andanother—statues—in the second conjunct. Assume that thepiece of clay \(c\) denoted by \(A\) in the first conjunctconstitutes, at time \(t\), the statue \(s.\) Then assuming thatstatues are physical objects, there are two distinct physical objectsbelonging to different kinds occupying the same space at \(t\). Some,notably Wiggins (1980), hold that this is entirely possible: Distinctphysical objects may occupy the same space at the same time, providedthey belong to different kinds. The temporal parts doctrine supportsand encourages this view. A statue may be a temporal part of atemporally extended piece of clay. But one statue, it seems, cannot bea temporal part of another. Even so, however, the duality ofconstituter 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 \(B\) aredifferent 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 dualityof books and copies of books is unparsimonious and the relativist isnot committed to it. There is no reason to concede to the philosopherthat we do not actually purchase or readbooks; instead wepurchase and read onlycopies of books. Any copy of a book isjust as much the “book itself” as is any other copy. Anycopy of a book isthe same book as any other copy. NelsonGoodman once remarked that “Any accurate copy of the text of apoem or novel is as much the original work as any other”(Goodman 1968, p. 114). Goodman was not suggesting that thedistinction between poem and copy collapses. If it does collapse,however, we have an explanation of why any accurate copy is as muchthe original work as any other: any such copy is the same work as anyother.

Objection 5: Geach remarks that “[a]s for ourrecognizing relative identity predicables: any equivalence relation… can be used to specify a criterion of relativeidentity” (Geach 1972, p. 249). But §3 above contains acounterexample. Some equivalence relations are defined in terms of theI-predicable of a theory and hence cannot serve as such. (Any pair ofI-predicables for a fixed theory are equivalent.) In fact it seemsthat any equivalence relation presupposes identity (cf. McGinn 2000).For example, the relationx and y are the same colorpresupposes 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 toequivalence.

Reply: This is a good objection. It does seem toshow, as the objector says, that identity is logically prior toordinary similarity relations. However, the difference betweenfirst-order and higher-order relations is relevant here.Traditionally, similarity relations such asx and y are the samecolor have been represented, in the way indicated in theobjection, as higher-order relations involving identities betweenhigher-order objects (properties). Yet this treatment may not beinevitable. In Deutsch (1997), an attempt is made to treat similarityrelations of the form “\(x\) and \(y\) are the same \(F\)”(where \(F\) is adjectival) as primitive, first-order, purely logicalrelations (see also Williamson 1988). If successful, a first-ordertreatment of similarity would show that the impression that identityis prior to equivalence is merely a misimpression—due to theassumption that the usual higher-order account of similarity relationsis the only option.

Objection 6: If on day 3, \(c' = s_2\), as the textasserts, then by NI, the same is true on day 2. But the text alsoasserts that on day 2, \(c = s_2\); yet \(c \ne c'\). This isincoherent.

Reply: The term \(s_2\) is not an absolutely rigiddesignator and so NI does not apply.

Objection 7: The notion of relative identity isincoherent: “If a cat and one of its proper parts are one andthe same cat, what is the mass of that one cat?” (Burke 1994, p.138).

Reply: Young Oscar and Old Oscar are the same dog,but it makes no sense to ask: “What is the mass of that onedog”? Given the possibility of change, identical objects maydiffer in mass. On the relative identity account, that means thatdistinct logical objects that are the same \(F\) may differ inmass—and may differ with respect to a host of other propertiesas well. Oscar and Oscar⁻ are distinct physicalobjects, and therefore distinct logical objects. Distinct physicalobjects may differ in mass.

Objection 8: We can solve the paradox of 101Dalmatians by appeal to a notion of “almost-identity”(Lewis 1993). We can admit, in light of the “problem of themany” (Unger 1980), that the 101 dog parts are dogs, but we canalso affirm that the 101 dogs are not many; for they are “almostone”. 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 negligibledifferences can add up to one that is not negligible.

Reply: The difference between Oscar andOscar⁻ is not negligible and the two are notalmost-identical. Lewis concedes this point but proposes to combinealmost-identity with supervaluations to give a mixed solution to theparadox. The supervaluation solution starts from the assumption thatone and only one of the dog parts is a dog (and a Dalmatian, andOscar), but it doesn’t matter which. It doesn’t matterwhich because we haven’t decided as much, and we aren’tgoing to. Since it is true that any such decision renders one and onlyone dog part a dog, it is plain-true, i.e., supertrue, that there isone and only one dog in the picture. But it is not clear that thisapproach enjoys any advantage over that of relative identity; in fact,it seems to produce instances of RI. Compare: Fred’s bicycle hasa basket attached to it. Ordinarily, our discourse slides over thedifference between Fred’s bicycle with its basket attached andFred’s bicycle minus the basket. (In this respect, the case ofFred’s bicycle differs somewhat from that of Oscar andOscar⁻. We tend not to ignorethatdifference.) In particular, we don’t say that Fred has twobicycles even if we allow that Fred’s bicycle-minus is abicycle. Both relative identity and supervaluations validate thisintuition. However, both relative identity and supervaluations alsoaffirm that Fred’s bicycle and Fred’s bicycle-minus areabsolutely distinct objects. That is, the statement that Fred’sbicycle and Fred’s bicycle-minus are distinct is supertrue. Sothe supervaluation technique affirms both that Fred’s bicycleand Fred’s bicycle-minus are distinct objectsand thatthere is one and only one (relevant) bicycle. That is RI, or closeenough. The supervaluation approach is not so much an alternative torelative identity as a form of it.

Objection 9: One may argue that the plausibility ofRI rests to a large degree on certain linguistic phenomena and that wedo not need it after all because these phenomena are in fact apparentand the notion of relative identity can be explained away. Forexample, Moltmann (2013) has recently argued that some apparentstatements of sortal-relative identity can be analysed on linguisticgrounds as statements of absolute identity. Consider an example ofsuch statements: “This is the same lump of clay but not the samestatue as that”, which is uttered in front of, say, twophotographs: one presenting the clay and the other showing the statuemade out of the clay. Moltmann starts her analysis focussing on thetwo pronouns at stake:this andthat, the so-calledbare demonstratives. The analysis reveals two different functionsthereof: referential and presentational. Due to the referentialfunction a bare demonstrative picks up a unique feature, or a trope asMoltmann claims. Secondly, this feature (trope) is used to recogniseits bearer (in any possible world that is accessible from the worldwhere the pronoun is uttered). Now in the example abovethisrefers to a complex trope, say the complex composed of this particularbrownness and this roundness, and “with the help of” thistrope maps to a lump of clay as a bearer of this trope. Similarly,that refers to another complex trope, say this particularbrownness and this angularity and maps to a bearer of this trope. Thecrucial assumption here is that the tropes in question may havemultiple bearers, e.g., the brownness and roundness is a trope borneboth by the clay and the statue. Moltmann’s semantics of the(apparent) statements of sortal-relative identity of the form“This is the same … as that” has it that suchstatements are true ifthis refers to a trope such that atleast one, but not all, of its bearers is identical (in the sense ofabsolute identity) to a bearer of the trope to whichthatrefers. So our sentence “This is the same lump of clay but notthe same statue as that” is true when (i) one bearer of thethis trope is the same as a bearer of thethat tropeand (ii) another bearer of that first trope is not the same as abearer of the other trope.

Reply: The semanticsoutlined by Moltmann has its ontological cost because its proponentneeds to acknowledge that a single trope may have multiple bearers.But even if you are persuaded by her argumentation to thiscontroversial position, you may still accept RI. Moltmann’stheory concerns a specific class of relative identity statements anddoes not directly affect the applications of RI described above, whichneed not be rendered as identity statements involving baredemonstratives. For instance, accepting this theory you are at libertyto say that the original and remodeled ship are the same ship but notthe same collection of planks, whereas the reassembled ship is thesame collection of planks as the original but not the same ship. Soeven if the whole body of linguistic evidence for RI can be explainedaway, we may use relative identity as a theoretical explanatory tool,e.g., to provide resolutions to the paradoxes described above.

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The SEP editors would like to thank Christopher von Bülow forreporting a list of typographical and other issues in an earlierversion of this entry.