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How Composition Could be Identity

Paul Hovda

March 3, 2006

Abstract

The paper argues that, as far as logic and abstract metaphysics go,the claim that composition is identity could be true, despite a well-motivated argument against the coherence of the claim. The key toanswering the arguments is to reject a principle connecting identitywith plural expressions, one that has been assumed or ignored inmost recent philosophical discussions of the logic, semantics, andmetaphysics of plurals. We call it the principle of the NumericalTransparency of Identity (NTI). It says that for any plural terms tand s, the sentence pt are identical with sq (interpreted collectivelyat both nouns, not distributively) is logically equivalent with the sen-tence pEvery one of t is one of s and every one of s is one of tq.

We show that the denial of NTI is logically coherent. The keysteps of the case against NTI include an analysis of collective anddistributive plural predication that motivates our claim that ‘is oneof’ does not express a relation; a picture of the metaphysics of pluralproperties and relations that motivates our view of collective pluralidentity predications; and a clear semantic theory for a language inwhich NTI fails, which meshes with the analysis and the metaphysi-cal picture.

Contents

1 The Question 11.1 The numerical transparency of identity . . . . . . . . . . . . 81.2 Composition as identity and mereology . . . . . . . . . . . . 101.3 NTI and contemporary formalisms of plurals . . . . . . . . . 111.4 Scope of the thesis that composition is identity . . . . . . . . 13

2 ‘is one of’ does not express a relation 14

3 Plurals, properties, and relations 23

4 Semantics 30

5 Glimpses Beyond 335.1 Restricted NTI and mereological notions . . . . . . . . . . . . 345.2 For any things, a thing . . . . . . . . . . . . . . . . . . . . . . 365.3 Composition as identity . . . . . . . . . . . . . . . . . . . . . 395.4 Singularities and RNTI . . . . . . . . . . . . . . . . . . . . . . 40

How Composition Could be Identity 1

1 The Question1

In Parts of Classes, David Lewis praised mereology for its “ontologicalinnocence” as well as for its intuitiveness and ability (with certain assump-tions in place) to give a kind of metaphysical foundation for (the less intu-itive, in his view) set theory: ontologically commit yourself to some thingsone by one, or commit yourself to their mereological fusion; it’s the samecommitment either way. Thus, commitment to a fusion of some thingsis no further commitment than commitment to each of them. This claimabout ontological commitment seems to require an amazing metaphysicalclaim: for the many to compose the one fusion is just for the many to be (inthe sense of identity) the one fusion.

There is something attractive and plausible in this idea, but seriousproblems loom. Lewis himself was quick to back off of a literal readingof the identity claim, and retreat to the position that the special relationbetween the many and the one, namely composition, is not identity itself,but rather something very much like it.

But if we give up the claim that composition is identity, it is not clearthat we can keep the claim that mereology is ontologically innocent. Set-ting aside Lewis’ other theoretical jobs for mereology, I believe that thereis more logical room for the view that composition is identity than Lewissees.

The bulk of this paper will answer an argument that may be implicit inLewis’ remarks, but is given a more focussed statement in other authors.Before proceeding, however, it will be helpful to sketch an application ofour general outlook, an application that shows why mereology is indeedontologically innocent.

Suppose that Alfred ontologically commits himself to Tom (whom Al-fred also knows to be a cat), and to Jerry (known to be a mouse). SupposeAlfred also knows that no cat has a mouse as a part, and no mouse hasa cat as a part. Then the mereological fusion “Genie”, of Tom and Jerryis neither a cat nor a mouse, since Genie has Tom as a part and has Jerryas a part. As Byeong-Uk Yi argues in [17], it appears that the fusion Ge-

1Acknowledgements: This paper has benefited from conversations between its authorand Mark Bedau, Ben Caplan, Eddie Cushman, Louis deRosset, Mark Hinchliff, DavidKaplan, D. Anthony Martin, Rotem Rabinovich, and Gabriel Uzquiano.

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nie is a genuinely new thing, and there is no obvious reason why Alfred’sstated commitments should also commit him to it. Why should we thinkthat ontological commitment to Tom and to Jerry already is commitment tosomething that is neither a cat nor a mouse?

We proceed in stages. First, we claim that Alfred is committed to threedifferent ways that the property of weighing ten pounds could be instanti-ated: it could be had by Tom; it could be had by Jerry; and it could be hadby Tom and Jerry collectively. These are three logically distinct possibili-ties; call them P1, P2, and P3.2 Note that P3 is not a truth-function of P1 andP2, but a genuinely distinct possibility. Similar remarks go for other prop-erties, and for relations, too. Second, we claim that P3 may be expressedin two different ways: first, by plural reference to (each of) Tom and Jerry,which Alfred may achieve by ‘Tom and Jerry’, followed by collective pred-ication of the property in question; second, by singular reference to Genie,followed by (singular) predication of the property in question. Third, weclaim that it is sufficient for ‘Genie’ to refer to Genie if we stipulate: Let‘Genie’ refer to Tom and Jerry (collectively). Just as commitment to Tom andto Jerry yielded three ways that weighing ten pounds could be instantiated,it also yielded three ways that ‘Genie’ refers to could be instantiated. Ourstipulation makes ‘Genie’ bear the reference relation in the third way, toTom and Jerry collectively. It refers to Tom and Jerry, but not to either ofthem. (Contrast ‘Tom and Jerry’, which refers to Tom and refers to Jerry.)Fourth, we note that whether or not Alfred chooses to make such a stip-ulation, his commitments make it available to him. Thus, unless his singularquantifiers are for some reason stipulated to be restricted, e.g., so that theyrange only over cats and mice, he may truly say “There is something thatis neither a cat nor a mouse”.3 (Again, the property of being a cat and theproperty of being a mouse will each be instantiable in three ways, and sim-ilarly for the property of being neither a cat nor a mouse.) Speaking with his

2The idea that there are three possibilities here is used to the effect that plural quantifi-cation is not innocent of commitment to abstract objects like sets (pace Boolos), by AllenHazen [2]. Hazen’s point seems to me to be at the heart of the matter, though we our useof it is quite different from his.

3To connect quantification with the notion of a bound variable, and truth-conditionswith variable assignments: in Alfred’s situation, he is committed to the possibility of asingular variable bearing the is assigned to relation to Tom and Jerry (while not bearing itto either of them).

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quantifiers wide open, Alfred may truly utter this sentence. Finally, wenote that if Alfred were to protest that he can assure us that he is not on-tologically committed to a single thing that is neither a cat nor a mouse, wewould reply that he may be right, but that we have here the makings onlyfor a verbal dispute about how to express his commitments. If by “singlething” he means “either a cat or a mouse”, then yes, he is not committedto such a thing. If he means “thing that falls under a natural kind”, thenprobably he is right. But if he means “possible objectual component of afact”, then he is wrong.

Commitment to Tom and to Jerry turned out to be commitment also toGenie. Genie is, in one sense, another thing: it is not identical with Tom,and it is not identical with Jerry. But in another sense it is not, since itis identical with Tom and Jerry. We now note that (for all we have said)Alfred is not committed to a fourth thing, not identical with any of Tom,Jerry, or Genie. It was crucial to our argument that for Tom and Jerrycollectively to bear a property is not for either one of them to bear it. Thiswas guaranteed by the joint truth of the facts that Tom is a cat, that Jerryis a mouse, that no cat is part of a mouse, and that no mouse is part of acat. But for Tom, Jerry, and Genie collectively to bear a property is nothingmore or less than for Genie to bear it. This is because, again, Genie isidentical with Tom and Jerry (collectively), and hence Genie is identicalwith Tom and Genie (collectively), and hence with Tom, Jerry, and Genie.In connection with this we note that, since composition is identity, therelation of part to whole is ultimately to be understood in terms of identity:what it is for Tom to be part of Genie is for Genie to be identical with Tomand Genie (collectively).

There are many objections that might be made to our argument aboutAlfred’s commitment. It is plausible that there is a sense of “ontologi-cal commitment” that is highly deferential to the subject’s belief system,so that one might be committed to Hesperus without being committed toPhosphorus; or committed to water without being committed to hydro-gen; or committed to Tom without being committed to Tom’s brain or toany carbon atoms; or committed to Tom’s atoms without (despite the factthat they are arranged, as they are, “cat-wise”, in their current environ-ment with its actual history) being committed to Tom. But surely thereis another sense of commitment, (and, if you like, of “what your quanti-fiers range over”) on which none of these pairs of commitment and non-commitment are possible, even though the subject may not know it, may

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not be able to know it a priori, indeed may not be able to know it at all.Many of the possible objections can be met simply by making this distinc-tion.

Another kind of objection can be made: Someone we might call a “sin-gularist mereologist” might accept that Alfred is committed to Genie (inthe relevant sense of commitment) but reject our account of why this is.He might suggest that formal mereology is a body of necessary truths, andthat this is why commitment to Tom and Jerry necessitates commitment totheir fusion, Genie. We agree; the disagreement is over our suggestions,in particular, that it is sufficient to refer to Genie that one refer to Tom andJerry collectively, and, in general, (roughly expressed) that for Genie to en-ter into a fact is no more or less than for Tom and Jerry collectively to enterinto it. The “singularist” will insist that it is illegitimate to regard Genie asanything but a single object to which Tom and Jerry each bear the primi-tive is part of relation. Our dispute is thus about whether the singularistnotions of part, composition, and fusion can be explained in terms of iden-tity and plurality. Our defense of this general idea is spread throughoutthe rest of the paper.

Attractive as the thesis that composition is identity might seem, thereis a logical difficulty at the very heart of the matter. Here is an argumentthat shows it quite clearly, adapted from Yi (from [17]).4

An adaptation of Yi’s argument goes like this:(0) Tom, Jerry, and Fido are pair-wise dis-

tinct.Premise

(1) Tom and Jerry compose Genie. Premise(2) Composition is identity. Suppose for reductio(3) Tom and Jerry = Genie From (1) and (2)(4) Tom is one of Tom and Jerry and Fido. (logical truth)(5) Tom is one of Genie and Fido. From (3) and (4)(6) Fido is neither a cat nor a mouse. Premise(7) Genie is neither a cat nor a mouse. Premise(8) Every one of Genie and Fido is neither a

cat nor a mouse.From (6) and (7)

(9) Tom is neither a cat nor a mouse. From (5) and (8).

4Arguments to similar effect are found in Sider [10]. Sider focusses on plural quan-tification, and plural variables. Lewis himself makes some remarks suggestive of thisargument [5].

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This argument has an unacceptable conclusion, and the only suspiciouspremise is the thesis that composition is identity. Thus the proponent ofcomposition as identity must explain how the reasoning goes wrong. Themistake, I argue, is the assumption of a principle that would validate theinference from (4) to (5). I accept all the other steps in the argument. Onemight try quarreling with (8), exactly on the grounds that (5) is true! Thisseems to me to be a mistake. I take it to be a logical truth that: Every oneof Genie and Fido is either identical with Genie or identical with Fido. (5)is false.

The principle needed to make the argument go through, and which Ireject, I call the Numerical Transparency of Identity (NTI). NTI says that anyconstruction of the form

➀ (collectively) are identical with ➁ (collectively)

is logically equivalent with the corresponding instance of

for all things, that thing is one of ➀ if and only if it is one of ➁

where the substituends for ➀ and ➁ are appropriate—roughly, are “refer-ential” plural noun phrases like ‘John and Paul’, ‘they’, and, perhaps, ‘thestudents’. (Referential, as opposed to quantificational, as ‘all men’ and‘some men’ are.) The aim of this paper is to argue for a coherent pictureon which NTI fails.5 We will discuss the name of the principle, and whatcount as “appropriate” substituends in more detail, momentarily. First, alittle more background.

There are two logical aspects of plural constructions: distributive andcollective predications involving plural nouns; both aspects are hard to fitinto the straightjacket of first-order logic (and thus many of us will be onrelatively unfamiliar ground when we think about them). A famous ex-ample of a (non-first-order) distributive use is the Geach-Kaplan sentence:‘Some critics admire only one another.’6 This is distributive because of the

5It may be worth remarking that I do not find these arguments conclusive, thoughI do tend to find the arguments against NTI more compelling than the arguments forit. My intention is to raise the question whether NTI is correct, and advance seriousconsiderations against it, and thus, at least to initiate discussion of the issue.

6For the argument that this sentence has no first-order formalization, not even an in-direct one, see [1], especially p. 433.

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phrase ‘one another’; roughly, the admires relation is distributed (in a par-ticular way) over the critics. An example of a collective use is ‘John andPaul (together) lifted the piano’. This is equivalent to no truth-function of‘John lifted the piano’ and ‘Paul lifted the piano’, and, hence, cannot bestraightforwardly given a first-order treatment.

NTI is about the connection between collective predication of identityand a certain distributive predication. It is difficult, at first, to grasp justwhat NTI says, because we are not used to predicating identity collectivelyplurally. But, on the face of it, it is possible to do so (pace Peter van Inwa-gen).7 We can make sense of the sentence

John and Paul are heavier than George.

The sentence

John and Paul are identical with George.

has exactly the same (surface-)grammatical form. We can easily grasp thecollective and distributive readings of the first, and we can easily grasp thedistributive reading of the second: it simply says what is said by ‘John isidentical with George and Paul is identical with George’. (Or, if it doesn’tsay exactly the same thing, it says something logically equivalent withthis.) The collective reading of the identity sentence is harder, but you getused to it after a while. For example, there are fairly obviously true collec-tive identity sentences, like ‘John and Paul (collectively) are identical withJohn and Paul (collectively)’ and ‘John and Paul (collectively) are iden-tical with Paul and John (collectively)’. These appear to have the samegrammatical form as the readily intelligible ‘John and Paul (collectively)are heavier than George and Ringo (collectively)’.

Composition as identity motivates the rejection of NTI; but there isa serious argument for NTI (and hence, against composition as identity)that must be addressed. NTI is derivable from a substitutivity priniciplegoverning plural expressions, reminiscent of the familiar “substitutivityof identicals” for singular expressions. Suppose we knew that collectiveidentity supports substition for any context φ; that is,

7In [15], van Inwagen asks: “what kind of syntactical sense is ther in taking either‘is’ or ‘are’ and putting a singular term or variable on one side of it and a plural term orvariable on the other?” (p. 211 of [15]) He then argues that no answer can be given. Hisarguments are various, but they can all be answered in the framework advocated here.

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logically implies

φ(➀) if and only if φ(➁)

where φ(➁) arises from φ(➀) by substituting one or more occurrences of➀ in φ(➀) with ➁. Then we would know that NTI is true: we simply letφ(➀) be the indisputable

for all things, that thing is one of ➀ if and only if it is one of ➀

and substitute at the second occurrence of ➀.I will present three ideas about the logic and metaphysics of plural

constructions that will reveal why this kind of substitution might fail.Once this argument from substitution is rejected, NTI should be in seri-ous doubt. (For what else would motivate NTI?)

First, I will argue that ‘is one of’ does not function by expressing a re-lation. Thus, the failure of substitution here does not violate the law thatidenticals are indiscernible—that identicals have the same properties andbear the same relations. The argument flows from an analysis of the cen-tral kinds of plural predication. Second, I will give a metaphysical pictureof collective relation-bearing that shows how we should think of collectiveidentity and the kind of indiscernibility it entails. On this picture, thereis no fundamental distinction between plural and singular relations, andevery relation has a fixed arity that does not vary, no matter how manyobjects are involved in a given bearing of it. For example, the relation tolift is a two-place relation, even though any number of things (collectively)can lift a thing. Identity is a two-place relation, even when it relates somethings (collectively) to some things (collectively). But if it does relate somethings to some things like this, then any property had by the first things(collectively) will be had by the second things (collectively). Third, I willsketch a semantics for plural terms that will show how ‘is one of’ func-tions without expressing a relation. In brief, a plural term t refers to each ofsome things: pn is one of tq (with singular term n) will be true just in casethe object that n refers to is also referred to by t. The semantics, combinedwith the earlier arguments, should help to make clear how NTI can fail.

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1.1 The numerical transparency of identity

We must now get a little more particular about exactly what NTI says.NTI is intended to apply to singular terms as a limit case; let us be explicitabout this. A singular term as a substituend for ➀ in the form


results in an ungrammatical sentence of (American) English if we keep thewords exactly as written. So, as a limit case, we will count the followingas an instance of this form

John is identical with John and Paul (collectively).

When a singular term is a substituend for ➁, the ‘collectively’ is not neededor wanted, so we will also count both of these:

John and Paul (collectively) are identical with John.John is identical with John.

as instances of this form.It is also convenient to stretch English plurals to include singulars in a

couple of other ways. We will extend our conception of the form

for all things, that thing is one of ➀ if and only if it is one of ➁

similarly, except that for this, we do not need to change any words in theresult of the substitution; we need only to ensure that the result of substi-tuting singular terms (for either ➀ or ➁) is recognized as grammatical bythe reader. Accordingly, in the rest of the document, I will assume that itis true, for any singular term s

For any thing: it is one of s if and only if it is identical with s

I will also assume that when we say “for any thingsxx, . . . ” we mean toinclude, as a limit, that there is exactly one thing that is one of themxx.(Just as we may use subscripted singular first-order variables on singularEnglish nouns to indicate anaphoric connections, we will use subscripted“plural variables”, as above, to indicate anaphoric connections among plu-ral nouns.)8 This usage may not be standard English, but it is generally

8As of yet, there is no standard way of writing formal plural variables. Some alterna-tives include

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followed in the contemporary study of the logic of plurals. Note that withthis usage, the following is a logical truth:9

For any thing x, there are thingsxx, such that for every thing y:y is one of themxx if and only if y = x.

NTI is a principle saying that two forms are logically equivalent. Cor-responding to it is the sentence that represents its universal generalization.Let us call this sentence,

For any thingsxx and any thingsyy:theyxx are (collectively) identical with themyy (collectively)

if and only iffor any thing, it is one of themxx if and only if it is one of themyy.

Material NTI.NTI says that two forms are equivalent. I believe that one of these

forms is strictly weaker than the other. The doubtful half of NTI is the onethat says that


logically implies

for all things, it is one of ➀ if and only if it is one of ➁

It will be useful to have short labels for these forms. Let us say that thefirst proposes the collective identity of ➀ with ➁. Let us say that the secondproposes the distributive identity of ➀ with ➁. (But please keep in mind that‘distributively identical’ does not express a relation—it is a contextuallyeliminable abbreviation.) I am suggesting that collective identity does notimply distributive identity. I see no reason to deny the other half of NTI: Iaccept that distributive identity implies collective identity.

The principle is called the “Numerical Transparency” of Identity be-cause it tells us, in effect, that the collective identity of n things with m

x, X, xs, Xs, the Xs, xx

We generally adopt the last, but will often stick close to English by using them as sub-scripts on English plural nouns.

9We will sometimes harmlessly conflate ‘is identical with’ and ‘=’.

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things entails that n = m. Thus, for any things, there is a unique num-ber associated with them, no matter how they are described. If NTI fails,then some n things might be collectively identical with some m things,even though n 6= m. In that case, the distinct numbers n and m are equallyapplicable to them. Thus, to know that they are n in number may be com-patible with their being m in number, and one may or may not know thelatter; thus, their being m in number (or not) is not transparent to one whoknows that they are n in number, in the way it would be if NTI held. Itwill emerge, on our view, that there is a unique number associated witheach plural referring expression, while there is no guarantee that this num-ber uniquely fits the things, the purely worldly component of “what isreferred to by the expression”. Thus, there is a unique “referential” num-ber, namely two, associated with ‘John and Paul’, but there may be nounique number associated with John and Paul. Now, if John and Paul arenon-identical and are mereological atoms—that is, if each has no parts inthe sense of part captured by formal mereology and definable in terms ofidentity—then there is a special metaphysical number for them: two. Ingeneral, if some things are made entirely of mereological atoms, there willbe associated with them a unique “metaphysical” number (of their atoms),and if each of them is an atom, this metaphysical number will be identicalwith the referential number associated with any term that refers to each ofthem (and to nothing else). But things that have any atomless parts willnot even have a unique number in this way, and if one of some things isnot an atom, then their metaphysical number (if it exists) will be strictlygreater than the referential number of any term that refers to each of them(and to nothing else).

1.2 Composition as identity and mereology

We have seen that if NTI is correct, composition is not identity. We cansee also that if NTI is incorrect, then it looks like identity is, or is one kindof, composition. Suppose NTI fails. Then there can be some things (callthem “the Xs”) (collectively) identical with some things (call them “theYs”) and some thing x that is one of the Xs and not one of the Ys. x isnot one of the Ys, but it is intimately connected with the Ys, since it is oneof some things that are identical with the Ys. One might say “its identityoverlaps the identities of the Ys, and is somehow entirely overlapped”.

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The connection is perhaps clearer in the special case in which there is onlyone thing y that is one of the Ys; i.e., every thing that is one of the Ys isidentical with y. (To allow that there is only one thing that is one of “somethings” stretches English, of course, but recall that, for convenience, weallow it in this discussion.) Then the Xs are (collectively) identical withy. Thus x, though it is not identical with y, is one of some things thatare (collectively) (identical with) y. The relation between x and y is notidentity, but it is importantly connected to identity. One might say “theidentity of x is not entirely separate from the identity of y,” or even thatthe identity of x is part of the identity of y. One might plausibly suggest, inmore familiar vocabulary, that this intimate connection between x and y isthe relation of part to whole, or is a special kind of part-to-whole relation:x is part of y.

As we suggested earlier, we take it that part of the content of the thesisthat composition is identity is that the (mereological) notion of part is tobe understood in terms of identity, as just suggested.10

1.3 NTI and contemporary formalisms of plurals

NTI may be regarded as a putative logical principle, and so we shouldask how it has been treated in contemporary discussions of the logic andsemantics of plurals. It turns out that it is not directly discussed. This ispartly because there seems to be no consensus about how to think of therelationship between plural and collective plural predication.11 In threerecent studies, however, ones rich enough to cover both distributive andcollective predication, NTI is not directly discussed, but something thatmight look like it is. (I have in mind the discussions of Yi [16], McKay[6], and Rayo [8] .) All three studies consider formalized plural languages,and in these they define a plural “identity” predicate ≈, basically as an

10Ted Sider takes a different approach when we gives a limited defense of the claimthat composition is identity in [10]. He takes the notion of composition to be defined interms of the notion of part, and then takes the claim that composition is identity to bethe claim that some things compose a thing only if (and if) they are identical with it. Thenotion of part is left as a primitive, if I understand him right.

11Boolos, for example, did not deal with collective predication (or does not make thedistinction), which makes sense, given his motivations.

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abbreviation.12 NTI holds by definition for ≈.The effect (achieved in slightly different ways by different authors) is

to define the plural “identity” predicate in terms of “is one of”. Let us use‘x’ and ‘y’ as formalized singular variables, and ‘xx’ and ‘yy’ as formalizedplural variables, and ‘OneOf’ as a formal symbol with the syntax of a two-place relation, the first place of which admits formal singular terms, thesecond place of which admits formal plural terms. Yi directly defines

xx ≈ yy

as

∀x(OneOf(x, xx) ↔ OneOf(x, yy))

This is the formalized version of

For any thing, it is one of xx if and only if it is one of yy.

Rayo effectively makes exactly the same definition (but within a muchlarger framework). McKay’s treatment is slightly different, but basicallyhas the same effect.13

It might thus appear that these authors take NTI and Material NTI tobe “true by definition.” But this is not, I think, a correct description oftheir stated views: NTI is not a principle about a defined predicate ≈.Material NTI does not contain a defined predicate. Material NTI, and NTI,are about the identity relation itself.

Anyone with views like those of these authors might, nonetheless, sug-gest that Material NTI is itself an analytic truth, or at least some kind oflogical truth, connecting ‘identical with’, in its collective plural use, with‘is one of’.

12See section 4 of [16], chapter 6 of [6] and the Appendix of [8].

13He effectively defines singular variables in terms of plural variables and Among, sothat he does not need or have OneOf as a separate predicate. (This is what he does inchapter 6 of [6], at least; in chapter 3 his syntax is more like Yi’s.) He writes ‘xx ≈ yy’ asan abbreviation of ‘Among(xx, yy) ∧ Among(yy, xx)’ His definitions make it clear that, as adefined form, ‘∀x(OneOf(x, xx) ↔ OneOf(x, yy))’ is equivalent with this.

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1.4 Scope of the thesis that composition is identity

There are two ways one could take the thesis that composition is identity:it could be the all-encompassing thesis that whenever composition occurs,it is identity; or it could be the more modest claim that identity is one kindof composition. I myself find the latter view much more plausible. As Iwill suggest later, it is plausible that classical mereology is a correct formaltheory for the kind of composition that is identity. One who thinks thatmereology is the one correct general theory of the one relation of compo-sition (and the attendant relation of part to whole), as David Lewis did,should, according to me, think that (all) composition is identity.14 Onewho merely thinks that there are mereological fusions, and that there isa domain of objects with a part/whole relation that behaves, on that do-main, as classical mereology requires, should accept that that kind of com-position is identity. One may then also hold that there are other kinds ofcomposition and parthood: e.g., the relation between a cat and its tail.15

14The compositional nihilist also might reject NTI, perhaps with some qualifications. Thecompositional nihilist holds that it never happens that some things compose somethingelse. (Peter van Inwagen entertains the position, under the name ‘Nihilism’, but does notadvocate it, in Material Beings. Rosen and Dorr argue that it is a live possibility in [9].)Now, it depends how this is meant, but it might be that the nihilist will allow that therebe a thing that is identical with some things (collectively). If he allows that the atoms (theones that “make up” a human being) collectively have the property of singing, or beinghuman-shaped, or being human, then why not allow that there is something (but notsomething distinct from them!) that has these properties? If it can be shown that talk likethis does not carry any objectionable further ontological commitment, then why not allowit? (As we will explain in section 5.4, given certain reasonable assumptions, the nihilistwill be able to systematically “translate” such talk into talk in which quantifiers rangeonly over mereological atoms. This might be taken to “cash in” the apparent ontologicalcommitment to things other than atoms.)

15I believe that there is a special kind of part/whole relation—call it “organic part”—that holds between a cat and its tail, or between a cat and a water molecule that it has in-gested, but not between a cat and a steel ball that it has swallowed, nor between a cat andsome shrapnel lodged in it. Organic part is not the same as the formal part/whole rela-tion of classical mereology. There are well-know arguments against the claim that a cat isa mere mereological fusion of its parts, turning on temporal and modal considerations—living things grow and diminish, and could have been made of different parts. An under-appreciated argument, I believe, is this: there is no explanation of the fact that the watermolecule is i-part of the i-fusion of the cat and the water molecule. But there is an ex-planation (in terms of life-processes) of why the water molecule counts as part of the cat(and why, for example, a steel ball that it has swallowed does not). If this or other such

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Here is an example that all might accept: When you play two notessimultaneously on a piano, middle C and the G a fifth above it, you play aharmonic interval, or dyad. (Think token not type.) There are at least threenumerically distinct sounds: the C, the G, and the dyad. If you can hearnormally, you can hear these three sounds. If your hearing is abnormal,you might only be able to hear the C, and not be able to hear the othertwo sounds. But it is impossible to hear the dyad without hearing the twonotes, and impossible to hear the two notes and yet not hear the dyad.(By “hear” I do not mean “consciously discern”.) Why is the one thingpossible while the others are impossible? Answer: because the two notesare the dyad. More precisely, the C and the G (collectively) are identicalwith the dyad. And, therefore, to hear the two notes is to hear the dyad.16

2 ‘is one of’ does not express a relation

We now turn to the first main consideration against NTI: we argue that ‘isone of’ does not express a genuine relation.

Leibniz’ Law and NTI

Recall that, in favor of NTI, the inference from collective to distributiveidentity might appear to be a special case of the “substituting of identi-cals”. But the substitutivity principle that would be invoked to yield NTIis a principle about the substitution of terms, not of things. We should notaccept that principle as a proper principle about identity itself and thingsthemselves.

Speaking of things, we should accept that: for any thingsxx and anythingsyy, if thesexx (collectively) are identical with thoseyy (collectively),then any property that thesexx (collectively) have is a property that thoseyy

arguments are correct, the notion of part answering to formal mereology is a differentnotion from the notion of organic part.

16This is not to say that what we call “attending to the dyad” and what we call “attend-ing to the two notes” are the same activity. What we mean by the latter requires attendingto each of the two notes, while what we mean by the former does not. Does this meanthat “attending to” is an intensional context? We will return to issues like this in section2; for now, we only want to motivate the question.

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(collectively) have. This is a proper, metaphysical, principle of the substi-tutivity of identicals.

Now we do not know that, for example,

John is one of

expresses a property had by (collectively) John and Paul. We do not knowthat

is one of

expresses a relation. If we did, we would have a good argument for NTI.But if ‘is one of’ does not express a relation, then the rejection of NTI doesnot violate the metaphysical thesis that identicals may be “substituted”.

Initial considerations

Consider what happens when we say

Each of John and Paul is a musician.

Is the effect of this to predicate a property, the property of being each amusician of John and Paul? Or is it to predicate a property, the propertyof being a musician, in such a way as to yield a proposition that is true justin case each of John and Paul has the property? The latter is a reasonableview. If so,

Each of is a musician.

does not express a property. Rather, when the blank is filled in, the resultexpresses what me might call a distributed proposition involving only theproperty of being a musician. The property is conjunctively or universallydistributed, with the effect that the result is true just in case each of Johnand Paul has the property.

I am here thinking of propositions as structured entities, roughly onthe model of Russell’s “singular propositions.” For obvious reasons, thatterm for them is potentially misleading, so I prefer to call them “objective”or “objectual” propositions. The point is that the objective propositionexpressed by

Each of John and Paul is a musician.

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plausibly does not contain the property of being each a musician, but onlythe property of being a musician. We have not said that it is the same objec-tive proposition as is expressed by

John is a musician and Paul is a musician.

That is one friendly idea; the other obvious one would be that the objectiveproposition expressed includes as its components each of: John, Paul, theproperty of being a musician, and a logical operation indicating the conjunc-tive (universal) distribution of this property over the former two objects.17

Now consider a parallel account ofOne of John and Paul is a musician.

The effect of this is not to predicate a property of being each a musicianbut, instead, to distribute the property of being a musician. This will be adifferent mode of distribution than in the example with ‘each of’; it will bedisjunctive rather than conjunctive.

One of is a musician.

does not express a property.Reconsider

John is one of .

I take it that this is equivalent withOne of is identical with John.

And we should say the same thing about it: it does not express a property;rather, when the blank is filled, the result is a distributed proposition. Itdistributes the property of being identical with John, and it distributes itdisjunctively.

We do not have here a conclusive argument that ‘is one of’ does notfunction by expressing a relation. We have only noted that it is natural toregard the proposition expressed by a sentence with ‘is one of’ in it as fail-ing to contain a relation that corresponds to that expression. Instead, the ‘isone of’ is associated with the ‘is’ of identity, and a logical operation of dis-junctive distribution, which either shows up in the objective propositionor somehow determines a disjunction in the objective proposition.

17Perhaps two objective propositions are “expressed,” the mere conjunction being themore metaphysically fundamental, arising after predication has been completed, but log-ically determined by the proposition in which the mode of predication appears as anelement. Cf. footnote 23.

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So what does ‘is one of’ do?

One might wonder how to describe the semantic function of ‘is one of’in giving truth-conditions, rather than when talking about propositionsexpressed by sentences. We will give a more detailed discussion of a truth-conditional semantics of plural language in section 4, but it will help tolook at one element of our theory now. The truth-conditional descriptionof the role of ‘is one of’ is not meant to compete with what we have saidabout objective propositions, but to provide a rather different angle onwhat we take to be, ultimately, the same basic idea about ‘is one of’.

We take a plural term (except in the limit case in which it is semanti-cally like a singular term) to refer to (each of) many things. This deservesemphasis; it is absolutely central to our view that plural terms are refer-entially quite different from singular terms. A singular term, on the otherhand, refers only once—there is exactly one thing to which it bears therefers to relation. We suggest that the truth-conditions for something of theform

s is one of t

(with singular term s and plural term t) are:

One of the things that t refers to is identical with the thing that s refersto; that is (since s is singular):there is something t refers to and which is identical with somethingthat s refers to.

Thus, the truth-conditions need not, and should not be given like this:

What s refers to bears the one of relation to what t refers to.

Our preferred truth-conditions ascribe a “meaning” (in a thin sense, atleast) to ‘one of’ without appealing to a relation at all. And it is worthnoting that on our view, “what t refers to” is a potentially misleading lo-cution. A plural term refers to (each of) many things. “what t refers to”should be compared with “what you own”.

Modes of plural predication

Let us reflect a little further on the idea that, roughly speaking, ‘one of’and ‘each of’ serve to indicate a particular way of distributing a property

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over some things. We will suggest that all ways of connecting a propertywith some things must be connected by one and only one mode of connec-tion, where these modes include the distributive modes and the collectivemode.

Consider the English sentence

John and Paul are heavier than Ringo.

This sentence is ambiguous. There are at least two readings, given, appar-ently unambiguously, by

John and Paul are each heavier than Ringo.

and

John and Paul collectively are heavier than Ringo.

But are these really unambiguous? Maybe they are each ambiguous inexactly the same way as the original. Consider the following “further dis-ambiguations”

John and Paul collectively are each heavier than Ringo.

and

John and Paul each are collectively heavier than Ringo.

(and the others that one can make in this way).Something has gone awry, one feels. These “further disambiguations”

are suspicious. Perhaps they are not even grammatical sentences. But ifyou thought that every occurrence of a plural noun phrase is susceptibleto at least two readings, then there would be no unambiguous sentencethat begins ‘John and Paul. . . ’. Similarly, if you thought that every occur-rence of a plural predicative expression is ambiguous, there would againbe no unambiguous sentence that begins ‘John and Paul. . . ’, for whateverfollows the dots would be ambiguous. Thus, we suggest, the ambiguity ofthe sentence is not due to there being more than one way to read ‘John andPaul’, nor is it due to there being more than one way to read ‘are heavierthan Ringo’. ‘John and Paul’ is semantically univocal, as is ‘are heavierthan Ringo’. (Or, the latter may as well be, for our purposes; we may ig-nore any senses of ‘heavier than’ other than the central one, if there areany.)

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The ambiguity of the sentence ‘John and Paul are heavier than Ringo’is due to there being more than one semantic connection that can be madebetween (the phrases)

John and Paul

and

heavier than Ringo

Let us suppose that ‘heavier than Ringo’ expresses a property; call it F.The question is: in what mode is F being predicated?

We suggest that there are at least three possibilities, in principle (ifnot in English). F might be disjunctively distributed, so that the result-ing proposition is logically equivalent with

John is F or Paul is F.

F might be conjunctively distributed, so that the resulting proposition islogically equivalent with

John is F and Paul is F.

Finally, F might be collectively predicated, so that the resulting proposi-tion is logically independent of any truth-function of

John is F

and

Paul is F.

The three propositions correspond to three English sentences:

One of John and Paul is heavier than Ringo.John and Paul are each heavier than Ringo.John and Paul collectively are heavier than Ringo.

Here are semi-formal schemes for the three English sentences, presentedin the same order:

F :: one of : John, Paul.F :: each of : John, Paul.F :: collectively : John, Paul.

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The semi-formal schemes are unambiguous representations of the threepropositions that result from the three possible modes of predication of F.

We may now diagnose the infelicity of the “further disambiguations”in English (e.g., ‘John and Paul collectively are each heavier than Ringo’).They do not make sense because these modes of predication cannot be mean-ingfully embedded or iterated. Further, ‘John and Paul each’ is not a noun-phrase on a par with ‘John and Paul’, to be treated semantically simplyin terms of its reference. Similarly for ‘John and Paul collectively’. Thesemantic difference between ‘John and Paul each’ and ‘John and Paul col-lectively’ is not to be thought of as like that of two noun phrases that referdifferently. The only referring is done by ‘John and Paul’; the more com-plex noun phrases represent the reference together with a logical opera-tion.

Yet, for ’John and Paul’ to be used as a subject for predication, somemode of predication must be attached. You cannot simply refer plurally andthen predicate a property. Again, ‘John and Paul are heavier than Ringo’ isambiguous not because ‘John and Paul’ is ambiguous, nor because ‘heav-ier than Ringo’ is ambiguous; there is no ambiguity about what property isbeing predicated. The ambiguity of the sentence is due to under-specificityabout the intended connection between the property and the objects re-ferred to. Thus we should never ask ourselves whether some things have agiven property or bear a given relation without having in mind a particularmode of plural predication.

If an expression that indicates a mode is added to a property-expression,as in ‘are collectively heavier than Ringo’ the complex whole, can be, ina sense, “predicated of” or, better, “asserted of” some things, withoutambiguity—but this operation includes two distinct logical components:the property to be predicated, and the mode of predication. These two com-ponents do not fuse to form another property. If they did, it would be am-biguous in which mode the resulting property was to be predicated. (Andthere would then seem to be no reason for the infelicitous “further disam-biguations” not to make sense.)

Thus, our theory about the English ‘is one of’ is this: it does not expressa relation, but is a combination of the ‘is’ of identity with an expression,‘one of’, that indicates what we have called a mode of predication, disjunc-tive distribution. Our semi-formal representation of ‘John is one of Johnand Paul’ is thus

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(John :: identical with) :: one of : John, Paul

‘each of’ and ‘one of’ are duals

It will help, to see through the appearance that ‘is one of’ expresses a re-lation, to observe that there is a systematic equivalence between sentencesinvolving ‘each of’ and sentences involving ‘one of’, very much like thatbetween ∀ and ∃.

Using our semi-formalism, for any property F

F :: one of : ➀

is equivalent with

NOT ( non-F :: each of : ➀ )

In English, we have the equivalence of the forms

One of ➀ is . . .

and

It is not the case that each of ➀ is not . . .

(Actually, this form is ambiguous: roughly, there is a scope ambiguity forthe second occurrence of ‘not’. We intend the narrow scope.)

Thus, the sentence

John is one of John and Paul.

which might be alleged to express the holding of a supposed is one of re-lation between John, on the one hand, and John and Paul on the other, isequivalent with

It is not the case that John is not identical with each of John and Paul.

or, to be clear that the “narrow-scope” reading is intended,

It is not the case that John is non-identical with each of John and Paul.

But surely ‘each of’, as used in this last sentence, does not functionby expressing a relation; this is further evidence, we suggest, that ‘is oneof’, in the original, does not either. One could hold, I suppose, that ‘isnon-identical with each of’ expresses a relation, and that this last sentenceexpresses the negation of the holding of this relation between John, on theone hand, and John and Paul on the other. Similarly, ‘loves each of’, as in

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John loves each of John and Paul

would express another relation, and the sentence would express the hold-ing of this relation between John, on the one hand, and John and Paul onthe other. But then the logical connection between this last sentence and

John loves John and John loves Paul

would depend on a special feature of the ‘loves each of’ relation. Since, forany verb V

John V each of John and Paul

is equivalent with

John V John and John V Paul

the current idea involves the postulation of a wealth of special relationswith special logical connections. It is much more plausible to separate theroles of ‘each of’ and the verb in the logical analysis of these sentences,and hold that ‘each of’ functions to express what we have called a modeof predication, connecting (the semantic value(s) of) the plural subject with(the semantic value(s) of) the predicate.

If we went so far as to say that ‘each of’ functions in such a way that

John loves each of John and Paul

expresses exactly the same proposition as

John loves John and John loves Paul

and if our semantics for that expression made clear how this could be thecase, then we would have an excellent explanation of the logical equiva-lence of these sentences, and for a host of other logical equivalences. Thenwe would make a parallel move for ‘one of’.

I am not sure, however, that the sentences of this pair express exactlythe same proposition. But I am sure that I can give a picture of the seman-tics for (formal correlates of) ‘each of’ and ‘one of’ that makes clear whythe sentences are logically equivalent, without the postulation of logicalconnections among properties. (I will fill out that sketch in section 4 af-ter the discussion of the metaphysics of properties and collective propertypossession.)

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3 Plurals, properties, and relations

We did not question the principle that identicals have all the same prop-erties, enter into all the same relations, and so forth. We questioned NTI,so we had to question the claim that ‘is one of’ expresses a property. Wefound good grounds to question it: ‘is one of’ is a composite of identityand ‘one of’, and the latter (along with its dual ‘each of’) expresses a modeof predication, and is not a relational expression.

We now motivate a picture of properties and relations and the meta-physics of collective property-bearing, and the mode of plural predicationwe called ‘collective’, that will help us to see what the appropriate pluralform of Leibniz’ Law really amounts to.

This in turn will allow us to answer the more general question: forwhat kind of context φ is the form

➀ (collectively) are identical with ➁ (collectively)∴ φ(➀) if and only if φ(➁)

valid?

Fixed arities for the “multigrade”

It is crucial to our conception that, in short, all properties and relationshave fixed “arities”, despite being able to admit more than one thing (si-multaneously) in an argument place when predicated collectively of manythings.

We take for granted that there are objects, properties, and relations thatsomehow come together to form (atomic) facts and propositions, and thatcomplex facts and propositions are somehow formed out of these. Wetake for granted that it is straightforward to see a systematic correlationbetween these entities and the sentences of a first-order language. Nowsuppose that “to lift” expresses a relation. The sentence

John and Paul lift the piano.

presents a logico-semantic problem, insofar as it does not exactly fit themold of any sentence in a first-order language. Whereas the (conjunc-tively) distributive reading of the plural predication reasonably can bethought of as expressing (when true) a conjunction of two atomic facts,the collective reading must be approached differently.

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One approach would be to think of ‘lift’ as here expressing a three-placerelation, that holds among John, Paul, and the piano. But then

John, Paul, and George (collectively) lift the piano.

would involve a different, four-place, relation, and

Some men (collectively) lift the piano.

would involve covert restricted quantification over relations; it would saysomething like

There are some men, and there are some relations, and each of thoserelations is a “lifting relation”, and those men bear one of those rela-tions to the piano.

Besides the implausibility of the suggestion of the covert quantificationover relations, this approach faces the problem of clarifying the notion ofa “lifting relation,” which would appear to be a new category of propertyof relations.

Perhaps ‘lifts’ expresses a single “multigrade” relation.18 A multigraderelation can apply as if it had any number of blank spots. But what exactlydoes this mean? Most attempts to make this out have focussed on formallogic and semantics, rather than metaphysics.19

I suggest that we think of

John and Paul (collectively) lift the piano.

as expressing (if true) the fact that the lifting relation holds between Johnand Paul (collectively, not each), on the one hand, and the piano, on theother. The lifting relation involved is the very same two-place relationthat is involved in the fact that John lifts the guitar. It is a “two-place”relation, but more than one thing can (simultaneously, so to speak) fill oneof its places. The collective mode of predication is what connects (one ofthe blank spots of) the relation with John and Paul.

This conception has a great advantage over conceptions on which whatis really going on in this case involves a three-place relation (or three-placeinstance or determinate of a multigrade relation). Consider the differencebetween (the collective readings of)

18The term seems to come from Leonard and Goodman [4].

19As in [7] and [12].

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John and Paul fight George and Ringo.

and

John and Paul and George fight Ringo.

If ‘fight’ in both examples acts as a four-place relation, it would seem thatthe very same proposition, one we might represent as

Fight(john, paul, george, ringo)

is being expressed by both sentences. Clearly, the two English sentencesare not logically equivalent. Thus it is much better to think of ‘fight’ asonce and for all expressing a two-place relation. We would then representthe propositions expressed by the two sentences as something more like

Fight(

john, paul︸︷︷︸ , george, ringo︸︷︷︸ )and

Fight(

john, paul, george︸︷︷︸ , ringo︸︷︷︸ )respectively.

This conception generalizes neatly: we may think of every propertyand relation has having a fixed finite “arity”, but as “accepting” any thingor any things (collectively) in any of its “blank spots”. This is not to say,of course, that you get a fact when you put some things (collectively) inthe one “blank spot” of any property; just to say that you get an (atomic,objectual) proposition, a “logically basic possibility of a fact, directly in-volving some objects”—at any rate, you get something of the same kindas what you get when you put one thing in each of the blank spots. (By an“atomic” proposition, I mean, roughly, one that involves no logical oper-ations like conjunction or quantification; nothing beyond predication, orpredication in a mode. An atomic proposition involves merely a prop-erty, or relation, and proposed bearers of it. By an “objectual” proposi-tion I mean roughly what is usually meant by a “singular” proposition—aproposition that directly involves objects; for obvious reasons, that term ispotentially misleading here.)

One upshot of our conception is that there is no fundamental meta-physical distinction between “plural” properties and “singular” ones. Theremay be, of course, properties that are actually possessed only by single

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things.20 But the propositions formed by predicating such a property ex-ist nonetheless—the point is that they are false. Similarly, there may beproperties that are actually possessed, but never by single things. Again,the propositions formed by predicating such a property of one thing willsimply be false, maybe necessarily false or even logically false, not non-existent.

We thus agree with Byeong-Uk Yi that some properties can accept manythings “as such” (that is, collectively) as arguments, and generally thatsome “blank spots” in relations can accept many things (collectively) asarguments. Unlike Yi, we think that this is so for all properties and rela-tions.21

Against the plurally plural

Another important upshot of our conception is that the “plurally plural”is metaphysically insignificant. Plurally plural talk is at best a mere verbalcode for plural talk, exactly because what it is for two “thingses” to havea property is either (1) for each of them to have it (in which case we havenothing new) or (2) for certain things collectively to have it; roughly theseare the things you get when you “put the two thingses together”. To beexact, call these things things+. Things+ are: the things such that (A) everyone of them is (either one of one of the first thingses or one of one of theother thingses) and (B) every thing that is (one of one of the one thingsesor one of one of the other thingses) is one of them.22

20Of course, we must be careful about the notion of a single thing. An absolutely sin-gular thing would be something that is not identical with some (more than one) things(collectively). (We discuss these objects, which we call singularities in more detail in sec-tion 5.1.) Perhaps there are such things; if so, it is plausible that there are properties thatare had only by a thing if it is in this way absolutely singular.

21See section 5 of [16].

22This is not to deny that there could be a term t, an allegedly plurally plural term, thatbears a relation (call it ‘reference’) to some things (collectively) and some other things(collectively) and maybe still other things. In fact, that is just how we think of gram-matically plural terms! The point is that any “property” that looks like a property ofthingses is just a property of things (a property that can be had by a thing or some thingscollectively) or definable in terms of plural quantification and properties of things.

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Identity

Identity is a relation, so it, too, can accept any number of things simulta-neously in either blank spot. But it is identity. So:

For any things xx and any things yy,if the result of putting xx (collectively)

in one blank spot of the identity relationand yy (collectively) in the other is true

then the result of putting xx (collectively)into the blank spot of any property,and the result of putting yy (collectively) into itwill have the same truth value.

A similar, more generalized form should hold for relations.23

It is important to note that the principle here is as much about the iden-tity relation as about substitution. Consider the position that says that thepseudo-relation we called “distributive identity” is in fact the identity re-lation, and that the plural relation we call “identity” is something short ofidentity—call it “sub-identity”. The advocate of this position might say:

Distributive identity guarantees substitution salva veritate in allcontexts; sub-identity does not. Therefore, distributive iden-tity deserves to be thought of as the plural identity relation,while sub-identity is a mere equivalence relation. If, as yousay, distributive identity is not actually a relation, then there isno plural identity relation.

We reply:

23A tricky question arises over the identities of objective propositions involving collec-tive plural predication. Suppose that xx are collectively identical with yy, but not dis-tributively identical. Then is the objective proposition expressed by “xx are collectivelyF” the same as the one expressed by “yy are collectively F”? It is tempting to distinguishtwo levels of objective propositions: on the less metaphysically fundamental level, be-fore the property is actually predicated, so to speak, we have something like 〈xx, coll, F〉,where coll indicates the mode of predication to be used, and each of xx is in the proposi-tion; on the more fundamental level, post-predication, we have something like 〈F, xx︸︷︷︸〉where F has been connected with xx collectively.

If this distinction is acceptable, then we may say that the pre-predication propositionsinvolving xx and yy are distinct, but the post-predication propositions are the same.

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Identity should be thought of first in terms of metaphysics, notlanguage. Given our way of thinking of the “multigrade”, wemay ask: Is there some relation such that of necessity: when itholds, of some things xx and some things yy, then the resultof putting xx into the blank spot of any property, and the re-sult of putting yy into it, will have the same truth value? Ifso, that is what deserves to be thought of as the identity rela-tion. Sub-identity is such a relation, and may as well be theonly such relation. There is no relation of “distributive iden-tity”. There may be something that looks grammatically likea relation-symbol, but it is a potentially misleading way of ex-pressing something analyzable in terms of ‘each of’, and hence,not a mere relation.

Failures of substitutivity and implicit distribution

Sometimes

➀ (collectively) are identical with ➁ (collectively)∴ φ(➀) if and only if φ(➁)

is invalid—when φ does not simply express collective predication of aproperty. ‘is one of’ creates such a context; what others?

Sometimes contexts seem to be “analyzable as” ones in which the “blankspot” is the operand for something including the mode-of-predication op-erators ‘one of’ or ‘each of’.

Some examples will convey the idea:

➀ are bandmates

is analyzable as

there is a band that ➀ each are in

➀ admire one another

is analyzable as

each of ➀ admires only other ones of ➀

Sometimes verbs seem to involve implicit quantification over events.In these cases, we may have contexts like this. E.g.,

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➀ conversed

may be analyzable as

there was a conversation, and each of ➀ partook in it

Thus we hold that many English (plural) predicates do not simply ex-press properties, but, rather, express operations involving ‘one of’ and‘each’ along with some properties. Many apparent failures of substitu-tivity of identicals (in the plural) arise from this fact.24 For example, if wethought that John is a fusion of molecules, we would not want to be ableto infer, from ‘John and Paul conversed’, that ‘Some molecules conversed’.The above analysis justifies the failure of this inference, even when allcomposition is identity and thus, John and Paul are identical with somemolecules.

Predicates like ‘conversed’ (as normally used) do not simply distributea property, the way ‘are cats’ distributes the property of being a cat. Butthey are not purely collective, either, as ‘weigh 100 pounds’ is when pred-icated collectively, for they do not simply express a property that might behad by one thing or by many things collectively. Instead they distribute arelation to an implicit object: ‘They converse’ says that each of them bearsthe is a party to relation to some one event.

For the thesis that composition is identity to be plausible, we seem torequire a three-way distinction between plural predications: those that arestraightforwardly distributive (as in ‘each of is a cat’, ‘are each a cat’ or‘are cats’), those that are relationally distributive (like ‘are class-mates’ and‘converse’), and those that are purely collective (like ‘weigh 100 pounds’when it is predicated collectively). Sometimes it is not clear in which cat-egory to put a given predicate: e.g., ‘ surround the building’. Our viewsuggests that the question is settled by whether the inference from

they surround the building

to

24I believe that this observation will greatly help the “nihilist” to respond to GabrielUzquiano’s arguments in [14]. Uzquiano argues that the nihilist—one who believes thatall things are (mereological) atoms cannot give plausible paraphrases of certain Englishsentences involving plural predicates and quantifiers. If the nihilist gives paraphrasesthat bring the ‘one of’ constructions to the fore, and then applies our semantics (below),he can meet Uzquiano’s challenge.

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their fusion surrounds the building

is valid. If so, it is a purely collective predicate; if not, then it is relationallydistributive. I tend to think it is ambiguous, and can be used to expresseither: (1) a collective and passive surrounding: a state they enter into “asone”, as a moat might surround a building; or (2) a distributed and activesurrounding: a state or event each of them enters into as a (partial) agent,as a team of soldiers might, following an elaborate plan, surround a build-ing. There may also be a collective and active sense, in which an anacondamight surround its prey; some humans might surround a building in thisway by accident. It is tempting now to reconsider what we said about tolift and to fight; they may be relationally distributive, rather than purelycollective.

4 Semantics

We have seen that it is a plausible claim that ‘is one of’ does not expressa relation, and that there is a reasonable picture of the metaphysics of col-lective property bearing that meshes with that claim. Building on theseideas, we now give the outlines of semantical theories for languages withplurals, in which Material NTI fails, and in which it is clear that ‘is one of’does not express a relation (and hence, anything analyzable in terms of itdoes not simply express a property or relation).

We do not assume Material NTI in our meta-language. We will be talk-ing about the “reference” of singular and of plural terms in our object-language, and we will collapse into this notion the idea of “reference” forvariables, relative to an assignment.

The crux of the theory is that singular terms refer only “once”, andplural terms refer “multiple times”. To be exact: For each singular term s,there is an object x such that for any things, s refers to (collectively) them ifand only if they are (collectively) identical with x. (This is a way of saying,with plurals, that for each singular term s, there is an object x such that: (1)s refers to x; and (2) for each thing s refers to, it is identical with x.) Note,however, that a singular term may “refer to more than one thing” in theloose sense that there be some things, more than one, such that it refers tothem (collectively). (This makes perfect sense on the assumption that NTIfails.)

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A plural term t may refer to more than one thing, in the strong sensethat it may refer to a thing x and to a thing y, with x not identical with y.A singular term cannot refer to more than one thing in this strong sense.Each plural term must refer to at least one thing, but there is no limit tohow many things it may refer to.

We now consider the semantic clauses for the crucial forms. First, theclause for ‘is one of’ will be:

A sentence of the form

s is one of t

is satisfied just in case the thing referred to by s is referred toby t.

(Note that if s refers to α, and α is identical with (collectively) β and γ, thens refers to (collectively) β and γ. This does not cause any trouble for thedescription “the thing referred to by s”. s does not refer to β, nor to γ; onlyto α—only to (collectively) β and γ.)

The clause for the collectively plural identity statements is

A sentence of the form

s are (collectively) identical with (collectively) t

is satisfied just in case the things referred to by s (collectively)are identical with (collectively) the things referred to by t.

Now suppose, for illustration, that someone simultaneously plays thenotes C, E, and G on the piano. We will take it that, for example

‘the C’ refers to c, that token of the note C‘the E’ refers to e, that token of the note E‘the G’ refers to g, that token of the note G‘the C and the G’ refers to c and refers to g‘the CG dyad’ refers to x, the thing identical with (collectively)c and g‘the EG dyad’ refers to y, the thing identical with (collectively)e and g‘the triad’ refers to z, the thing identical with (collectively) c, e,and g.

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Note that, for exercise, ‘the CG dyad’ does not refer to c, and that ‘the Cand the G’ does not refer to x.

Intuitively, the following are true:

The C is one of the C and the G.It is not the case that the C is one of the CG dyad.The C and the G (collectively) are identical with the CG dyad.The C, the E, and the G (collectively) are identical with (collectively)the CE dyad and the CG dyad.

Our semantic clauses assure this is the case.Turning to one-place predicates generally, we simply interpret them

with properties, so that a sentence of the form

s is F

(with s a singular term) is satisfied just in case the thing referred to by shas the property expressed by the predicate F.25 If t is a plural term, then

t are collectively F

will be satisfied just in case the things referred to by t collectively have theproperty expressed by F.

Distributive plural predication could be handled by forcing the sen-tence into a form so that ‘is one of’ is the only distributive predicate (as itis in most formalisms of plurals). Then the clauses for ‘is one of’ and forsingular predication would suffice. But we could also work directly andsay that

25It is perfectly possible to give a more “extensional” semantics for predicates, onwhich we do not appeal directly to properties. We would take predicates to “refer” toobjects, and give appropriate clauses for satisfaction. One-place predicates would havea reference relation formally the same as that for plural terms: they would refer manytimes over, each time to a thing (or, equivalently, to some things collectively). Two-placepredicates would require a three-place reference relation, relating the predicate manytimes over, each time to a thing (or some things collectively) and a thing (or some thingscollectively) in order. Though this kind of semantics does not interpret predicates withproperties, it requires some contentful assumptions about the availability and plenitudeof these reference relations.

Yet another semantics could be given, using sets in the interpretation of predicates. Butthe most natural ways to do this raise difficulties about the interaction of set theory withcollective identities, discussed below.

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each of t is F;

(with plural term t) is satisfied just in case: each thing that t refers to satis-fies ‘is F’. Similarly,

one of t is F;

(with plural term t) is satisfied just in case: some thing that t refers tosatisfies ‘is F’.

Two-place predicates will be interpreted by two-place relations, and soforth for predicates and properties of higher arity.

It follows from our semantics that for any plural terms tt and ss

tt = ss → (φ(tt) ↔ φ(ss))

(where ‘=’ expresses collective identity) is valid provided that tt does notoccur as an argument of ‘is one of’ (on the right), in φ. This correspondswith (indeed, follows from our semantics together with) the thesis thatidenticals share all their properties. This is the restricted form of the sub-stitutivity that we should expect.

The more exacting condition of distributive identity guarantees fullsubstitutivity of two plural terms.

∀x(OneOf(x, tt) ↔ OneOf(y, ss)) → (φ(tt) ↔ φ(ss))

is valid, for any φ. This unrestricted substitutivity follows from the factthat the truth of a “distributive identity” sentence requires that the twoplural terms involved be semantically identical: any thing the one refersto, the other refers to, and vice-versa; and there is nothing more to theseterms, semantically.

5 Glimpses Beyond

We have seen that NTI is not beyond question. We may hold that thecollective identity of some thingsxx with some thingsyy entails that ev-ery property (collectively) possessed by (or relation collectively born by)thesexx is also possessed by thoseyy, and yet does not entail that it notbe the case that there is something that is one of thesexx and not one ofthoseyy. ‘is one of’ does not express a relation that holds between a thing

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and some things (collectively). There is a reasonable semantical theory forplural languages that meshes with these claims.

We conclude that NTI is not a correct logical principle. This was themain argument of the paper. We will now consider a few issues that flowfrom this negative point and the theory of plurals we used to justify it.

5.1 Restricted NTI and mereological notions

There is a restricted version of NTI that I think we should accept. To statethe principle, Restricted NTI (RNTI), we will need some vocabulary thatwill be useful throughout the rest of the discussion, vocabulary that is sug-gestive of the link between collective identity and mereological notions.

Let us say that something x is a singularity if

for any things xx, if xx are (collectively) identical with x, thenevery one of xx is identical with x.

Let us say that some things xx are singularities if

every one of xx is a singularity

Let Material RNTI be the sentence

For any singularities xx and any singularities yy:xx are (collectively) identical with (collectively) yy

if and only iffor any thing, it is one of xx if and only if it is one of yy.

RNTI is then the general principle that correspondingly restricts NTI tosingularities.

I claim that RNTI is correct.This is not at all the trivial claim that NTI holds for things for which

NTI holds. In fact, the argument I like for Material RNTI is rather long.Material RNTI follows from two axioms about plural identity, togetherwith some very basic rules for plural logic, (which rules we do not havethe space to discuss in detail here).

These axioms are best stated with the help of some more useful defi-nitions. Let us say that a thing x is an identity-part of a thing y (i-part forshort) if

x and y are (collectively) identical with y

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Accordingly, say that a thing x is an i-part of (collectively) some things xxif

x and xx are (collectively) identical with (collectively) xx

(With this definition, it will follow, with some highly plausible assump-tions, that something is a singularity if and only if it is its only i-part.Singularities are their only i-parts. This would justify our calling them“logical atoms”, since i-part is defined in terms of the logical notion ofidentity—but that term has unwanted associations.) Say that a thing xand a thing y i-overlap if

there is something z such that z is an i-part of x and z is ani-part of y

We may now state the axioms that yield RNTI. The first axiom is

For any things xx and any thing x, if x is one of xx, then x is an i-partof (collectively) xx.

The second axiom is

For any things xx, and any thing xif xx are (collectively) identical with x and xx (collectively)then x i-overlaps one of xx.

Again, there is a long proof of Material RNTI from these axioms, togetherwith some very basic logical rules for plurals.

Material NTI is compatible with these two axioms. And Material NTIwould follow from them, along with further claim that for any things,each of them is a singularity. From my point of view, this further claim isthe fundamental metaphysical assumption that should be used to justifyNTI, if it is to be justified. But on the other hand, the fact that this furtherclaim entails Material NTI might be used as part of an error theory: itmight be suggested that philosophers who find NTI plausible are used tothinking about singularities, and in the domain of singularities, MaterialNTI is true.

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5.2 For any things, a thing

But if we believe that there are some non-singularities, the question arises:how many? The most natural suggestion seems to be the most extreme;we call it the axiom of Comprehensive Collective Identity (CCI). It is:

For any things, they are collectively identical with some thing.

Any things have a singular collective identity. This seems a fairly naturalassumption to make, once it is admitted that sometimes, some two or morethings are collectively identical with some one thing. One might suggestagainst this that some things have to be somehow unified to be collectivelyidentical with a single thing; but it is natural to reply that, if unity matters,we may, in effect, take co-existence to be a thin form of unity.

CCI does raise some technical issues. It tells us that there is a functionfrom the thingses, so to speak, to the things.26 And since it is identity, itis a one-one function: for any thingsxx and any thingsyy, and any thingx, if theyxx (collectively) are x and theyyy (collectively) are x, then theyxx(collectively) are (collectively) theyyy.

Does this not violate Cantor’s theorem? No, not for what we have said,and exactly because NTI fails. The fact that some things are not distribu-tively identical with some things is compatible with the first things’ beingassociated with (collectively identical with) the same single thing that thesecond things are associated with. Recall that some things xx are “dis-tributively identical with” some things yy just in case

every one of xx is one of yy, and every one of yy is one of xx

Again, from the fact that xx are (collectively) identical with (collectively)yy, it does not follow that xx are distributively identical with yy. This doesnot violate the indiscernibility of identicals, since distributive identity isnot a relation that can relate some things to some things, since it essentiallyinvolves ‘is one of’.

To be more precise about why Cantor’s theorem is not violated: If thethings form a set, the theorem tells us that there are strictly more subsetsof it than there are things. Now, for any subsets s and t, there are thingsxx that are all and only the members of s, and things yy similarly for t,

26But only so to speak: as explained in section 3, I do not believe there is any point,metaphysically, in plurally plural quantification.

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and s = t if and only if xx are distributively identical with yy. Say thattwo subsets are “i-equivalent” if the members of the one are collectivelyidentical with the members of the other. i-equivalence is an equivalence re-lation. There are exactly as many things as there are i-equivalence classes.Since there are i-equivalent non-identical subsets, CCI does not commit usto a violation of Cantor’s theorem. The theorem tells us that there mustbe rather many cases of things that are collectively identical but not distribu-tively identical with some things. (In a few paragraphs we say a little moreabout the structure of these i-equivalence classes.) This resolves the issueof cardinality.

There are further technical difficulties, however, involving set theory.The expression “the set that contains the things xx” is ambiguous; it couldmean “the set whose members are every one of xx and nothing else”. Butit could mean “the set that bears the membership relation to xx collectivelyand to nothing else.” The former is what we would usually mean: the setof dogs has each dog as a member, and has no other members. The latterset is a singleton, in the sense that it bears the membership relation onceand only once, if it exists. It is the singleton of the thing y such that xxcollectively are y, (if such a thing y exists).

CCI says that for any things xx, such a y does exist. But now we con-front issues with some formal resemblance to the cardinality issue above,and connected with Russell’s paradox. In fact, if every singleton is a sin-gularity, then it takes only the most basic further assumptions to get acontradiction.27

This might make us doubt CCI.First, it is worth noting that the same issue besets the axioms of Clasical

Mereology (CM), for in CM we have the very similar axiom that for anyobjects, there is a fusion of them (that is a single object). If every singletonis a mereological atom, then, with a fairly weak set theory, a contradiction

27Say that a singleton is “naughty” if there are some singletons that collectively bearthe membership relation to it, and it is not among them. The following comprehensionscheme is fundamental to the logic of plurals:

∃x φ(x) → ∃xx ∀x (φ(x) ↔ OneOf(x, xx))

If there is more than one singleton, then there is a naughty singleton. Let φ(x) be “x isa naughty singleton” and a contradiction ensues. It is crucial to the derivation of thecontradiction that Material NTI (with quantifiers restricted to the singletons) applies.

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ensues.28

Second, we are far from forced into a contradiction. It is not clear whatthe best response is, however. Perhaps not every singleton is a singularity.Perhaps there are unexpected relationships among sets whose membersare not singularities. At any rate, it appears that a safe attitude is this:set theory was formulated with singularities in mind, in singular logic. Ifwe restrict its formulation in our system so that something is a memberonly if it is a singularity, all should be fine. Pure set theory will not beaffected, if we suppose, as is plausible, that all pure sets are singularities.Only impure set theory requires reconsideration. It would be basicallyunaffected if the only non-singularities that are allowed to be members ofsets are ur-non-singularites: things that have no sets as i-parts.

One last technical matter should be addressed, connecting the failureof NTI with the idea of composition as identity. If we add another plau-sible axiom, the analog of what is called, in formal Mereology, the StrongSupplementation Principle (SSP), then we can tell that the i-part relationbehaves exactly as Classical Mereology (CM) says the ‘part’ relation be-haves.29 SSP says:

For any thing x, and any thing yif all i-parts of x i-overlap y, then x is an i-part of y.

In the resulting theory, to be a mereological atom is to be a singularity;for something to be the mereological fusion of some things is for it to beidentical with them (collectively), and so forth.

CM, formulated with plural quantification, says that the part-wholerelation on single things behaves structurally like the ≤ relation of a com-plete Boolean algebra, except that there is no 0 element. So we can now seewhat our axioms (if we include or can derive SSP) require for the structure

28Gabriel Uzquiano makes essentially the same point, and further points about thedifficulties of combining unrestricted mereology, set theory, and plurals in [13]. Uzquianois skeptical that the two full theories can be satisfactorily combined in full generality.

29Here is one way to formulate CM, using plural quantification. (This formulationtraces back to Tarski [11].) Adjust your notion of “part”, if necessary, so that everythingis part of itself. Say that two things ‘overlap’ if they have a common part. Say that athing x is the ‘fusion of’ some things if: every one of them is part of x and every part of xoverlaps one of them. We can now state the axioms as:(1) The part-relation is transitive;(2) For any things, there is a unique fusion of them.

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of the collective identity relation. For any things, there is the single thing(usually not a singularity) they are identical with, and the single things arearranged as elements in a complete Boolean algebra. For any things xx andany things yy, whether or not they are distributively identical, they are col-lectively identical just in case the least upper bound for xx (distributively)in the algebra—its mereological fusion—is the same as that for yy.30 Thistells us something about the structure of the i-equivalence classes definedin connection with Cantor’s theorem.

5.3 Composition as identity

So much for the technical issues raised by denying NTI. We now considerthe associated doctrine of composition as identity.

The notions of i-part, i-overlap, and so forth, defined above, appear tohave exactly the logical properties many philosophers would want to as-cribe to the part-whole relation (those axiomatized by Classical Mereol-ogy). This helps to justify the claim that collective identity is (a kind of)composition.

On one view, there is only one part-whole relation, and one kind ofcomposition. If collective identity is as we have said, the picture thatemerges is one on which the part-whole relation is a relation between athing (the part) and a thing (the whole) that can be given (defined) interms of identity: a part of a whole is a thing that is one of some thingsthat (collectively) are the whole.

On another view, there is more than one kind of composition; i-compositionwould be only one among them. What would another kind be? There aremany reasons for thinking that the i-part relation is different from the spe-cial relation that links a living thing with its parts; call it organic part. Th

how composition could be identity - reed collegepeople.reed.edu/~hovdap/hccbi0.pdf · alfred also...

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