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Identity and Plurals

Paul Hovda

February 6, 2006

Abstract

We challenge a principle connecting identity with plural expres-sions, one that has been assumed or ignored in most recent philo-sophical discussions of the logic, semantics, and metaphysics of plu-rals. We call it the principle of the Numerical Transparency of Iden-tity (NTI). It says that for any plural terms t and s, the sentence pt areidentical with sq (interpreted collectively at both nouns, not distribu-tively) is logically equivalent with the sentence pEvery one of t is oneof s and every one of s is one of tq.

There are reasons one might want to deny this principle. Forexample, one might hold that composition is identity: when somethings compose a thing, they are identical with it. But this is logicallycoherent only if NTI fails.

We show that the denial of NTI is logically coherent; hence NTIis not a correct logical principle. The key steps of the case againstNTI include an analysis of collective and distributive plural predi-cation that motivates our claim that ‘is one of’ does not express arelation; a picture of the metaphysics of plural properties and rela-tions that motivates our view of collective plural identity predica-tions; and a clear semantic theory for a language in which NTI fails,which meshes with the analysis and the metaphysical picture.

Contents

1 The Question 11.1 The numerical transparency of identity . . . . . . . . . . . . 51.2 Composition as identity and the failure of NTI . . . . . . . . 71.3 NTI and contemporary formalisms of plurals . . . . . . . . . 8

2 ‘is one of’ does not express a relation 9

3 Plurals, properties, and relations 19

4 Semantics 25

5 Glimpses Beyond 285.1 Restricted NTI and mereological notions . . . . . . . . . . . . 285.2 For any things, a thing . . . . . . . . . . . . . . . . . . . . . . 305.3 Composition as identity . . . . . . . . . . . . . . . . . . . . . 335.4 Singularities and RNTI . . . . . . . . . . . . . . . . . . . . . . 34

Identity and Plurals 1

1 The Question1

The aim of this paper is to give good arguments against a certain log-ical principle involving plural constructions, connecting distributive andcollective predications of identity.2 This principle seems to have been ig-nored, or tacitly assumed without careful consideration, in recent devel-opments in the project of formalizing the language and logic of plurals. Itsfailure seems to go hand in hand with the idea that composition, or at leastone kind of composition, is identity.

The principle, which I call the Numerical Transparency of Identity (NTI)says that any construction of the form

À (collectively) are identical with Á (collectively)

is logically equivalent with the corresponding instance of

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

where the substituends for À and Á are appropriate—roughly, are “ref-erential” plural noun phrases like ‘John and Paul’, ‘they’, and, perhaps,‘the students’. (Referential, as opposed to quantificational, as ‘all men’and ‘some men’ are.) We will discuss the name of the principle, and whatcount as “appropriate” substituends in more detail, momentarily. First, alittle background.

One of the limitations of standard first-order logic is that plural con-structions, and reasoning involving them, are not directly formalizable init.

There are two logically important aspects of plural constructions thatcannot be captured with the ideas of first-order logic: distributive and col-lective predications involving plural nouns. A famous example of a distrib-utive use that cannot be straightforwardly given a first-order treatment isthe Geach-Kaplan sentence: ‘Some critics admire only one another.’3 Thisis distributive because of the phrase ‘one another’; roughly, the admires re-lation is distributed (in a particular way) over the critics. An example of acollective use is ‘John and Paul (together) lifted the piano’. This is equiv-alent to no truth-function of ‘John lifted the piano’ and ‘Paul lifted thepiano’, and, hence, cannot be straightforwardly given a first-order treat-ment.

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For the most part, in the logical tradition following Frege, the limita-tion of first-order logic to singulars has not been seen as a problem: mostthings you might want to say with a plural construction can be re-cast insingular language. You have a surrogate for distributive uses if you cantalk about sets, which are single things that can bear the membership rela-tion to many things. Instead of distributing a property directly over manycritics, you can say: there is a set such that every member of it is a criticwho admires only other members of it. And you have a surrogate for col-lective uses if you can talk about mereological fusions, or some other kindof single things composed of many things. Instead of saying that John andPaul collectively lift something, you say that the fusion of John and Paullifted something. The distributive use of ‘John and Paul’ in this surro-gate can then be eliminated with a set. So plural constructions (and thusthe principles of reasoning involving them) have not been directly formal-ized, since they can be simulated, up to a point, with surrogate singularconstructions. The only obvious catch is that you need a bigger ontologythan the plurals seem to require.

Thus, as first-order logic and set theory were developed, there werefew attempts to give plural constructions a logical life of their own—nonethat caught on. There is not yet a standard way to extend the formaliza-tion we call “first-order logic” to accomodate plurals directly. But that ischanging; the last twenty years have seen a number of developments inthis direction, accompanied with suspicion about the singular surrogates;the work of George Boolos was perhaps the primary spur.4 Boolos’ worksuggested that the distributive uses of plurals are logically powerful andyet ontologically innocent: one can have the expressive power of the sur-rogate (sets with many members), without commitment to single objectsthat “encompass” the many. Boolos made less use of the collective usesof plurals; the work of Byeong-Uk Yi, among others, has gone some waytoward formalizing these uses as well.5 Boolos and Yi both offer goodcriticisms of the use of surrogates.

The topic of this paper is a principle that must be considered if we areto give a formalization for plural constructions in their own right. But Iknow of no explicit discussion of it; I believe I see it implicitly assumed incontemporary treatments, but it is often difficult to tell whether it is beingassumed or ignored. It is difficult to grasp what NTI says, because we arenot used to predicating identity collectively plurally. But, on the face of it,it is possible to do so. We can make sense of the sentence ‘John and Paul

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(collectively) are heavier than George’. The sentence ‘John and Paul (col-lectively) are identical with George’ seems to have exactly the same gram-matical form. There are fairly obviously true collective identity sentences,like ‘John and Paul (collectively) are identical with John and Paul (collec-tively)’ and ‘John and Paul (collectively) are identical with Paul and John(collectively)’. These appear to have the same grammatical form as thereadily intelligible ‘John and Paul (collectively) are heavier than Georgeand Ringo (collectively)’.

But why might one think NTI holds or fails? Against it, I believe thatthe best way to understand the thesis that composition is identity is as thethesis that the composers are (collectively) identical with the composed—but this view is coherent only if NTI fails.6 (We will see why below.)

The view that composition is identity is attractive for at least two rea-sons. First, it yields a kind of “ontological innocence,” of composed ob-jects: if the composed is literally identical with the composers, then on-tological commitment to the composed is not a further commitment be-yond commitment to the composers.7 Commit yourself to some things,and you, thereby, entitle yourself to all the things composed of them. Sec-ond, if composition is identity, then, perhaps, the logical properties of thepart/whole relation can be deduced from, or at least illuminated by, whatwe know about the logic of the identity relation. For example, the transi-tivity of parthood might be derived from the transitivity of identity.8

There are problems for the view that composition is identity that havenothing to do with NTI, but which turn on the temporal and modal dif-ferences between a typical composed object and its composers. If a livingbeing is composed of different parts at different times, then how could itbe identical with the parts it is composed of at one time? If it could havebeen composed of different parts, how could it be identical with the parts itis actually composed of? These problems can be addressed in a metaphys-ical framework like David Lewis’, with four-dimensionalism and counter-part theory. But even if we do not accept such a framework, and we rejectcomposition as identity for living beings or other things, it may still bethat there is a kind of composition (and an attendant kind of part/wholerelation) that is identity.

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 hear

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normally, 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.9

Composition as identity, whether once and for all, or only for one kindof composition, motivates the rejection of NTI;10 but there is a serious ar-gument for NTI (and hence, against composition as identity) that mustbe addressed. NTI is derivable from a substitutivity priniciple govern-ing plural expressions, reminiscent of the familiar “substitutivity of iden-ticals” for singular expressions. Suppose we knew that collective identitysupports substition for any context φ; that is,


logically implies

φ(À) if and only if φ(Á)

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

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

and substitute at the second occurrence of À.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, there

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is 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.

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 À 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 Á, 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 À if and only if it is one of Á

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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 À or Á) 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 amongplural nouns.)11 This usage may not be standard English, but it is gen-erally followed in the contemporary study of the logic of plurals. Notethat with this usage, the following is a logical truth:12

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 À if and only if it is one of Á

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It will be useful to have short labels for these forms. Let us say that thefirst proposes the collective identity of À with Á. Let us say that the secondproposes the distributive identity of À with Á. I am suggesting that collec-tive identity does not imply distributive identity. I see no reason to denythe other half of NTI: I accept that distributive identity implies collectiveidentity.

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

1.2 Composition as identity and the failure of NTI

Now that NTI is before us, let us see exactly why it is incompatible with thethesis that composition is identity. An example will illustrate the conflict.Suppose that all composition is identity, and that classical mereology iscorrect. Then, for any four things, there is a mereological fusion of them,that is identical with them. (Not identical with each of them, but identicalwith them collectively.) Let a, b, c, and d be four cats, and let z be theirmereological fusion. Let the fusion of a and b be x, and the fusion of c andd be y. Then, a, b, c and d compose z, and also x and y compose z. Sincecomposition is identity, a, b, c and d (collectively) are identical with x and y(collectively). This is a statement of collective identity. Now suppose thatNTI is correct. Then we can infer distributive identity from this collectiveidentity, so that each of x and y is one of a, b, c and d. But this is simplyfalse. Every one of a, b, c and d is a cat, but neither x nor y is a cat.13

Since NTI also applies to singulars, NTI also tells us that a, b, c, and d

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(collectively) are identical with z only if each of them is identical with z.This is not the case of course, for then each would be identical with eachof the others; e.g., a would be b.

So if NTI is correct, composition is not identity. We can see also thatif NTI is incorrect, then it looks like identity is, or is one kind of, com-position. Suppose NTI fails. Then there can be some things (call them“the Xs”) (collectively) identical with some things (call them “the Ys”) andsome thing x that is one of the Xs and not one of the Ys. x is not one ofthe Ys, but it is intimately connected with the Ys, since it is one of somethings that are identical with the Ys. One might say “its identity overlapsthe identities of the Ys, and is somehow entirely overlapped”. The connec-tion is perhaps clearer in the special case in which there is only one thingy that is one of the Ys; i.e., every thing that is one of the Ys is identicalwith y. (To allow that there is only one thing that is one of “some things”stretches English, of course, but recall that, for convenience, we allow itin this discussion.) Then the Xs are (collectively) identical with y. Thusx, though it is not identical with y, is one of some things that are (collec-tively) (identical with) y. The relation between x and y is not identity, butit is importantly connected to identity. One might say “the identity of x isnot entirely separate from the identity of y,” or even that the identity of xis part of the identity of y. One might plausibly suggest, in more familiarvocabulary, that this intimate connection between x and y is the relation ofpart to whole, or is a special kind of part-to-whole relation: x is part of y.

1.3 NTI and contemporary formalisms of plurals

In three representative contemporary discussions of the logic and seman-tics of plurals, ones rich enough to cover both distributive and collectivepredication, NTI is not directly discussed. (I have in mind the discussionsof Yi [17], McKay [8], and Rayo [10] .) Not directly discussed. All threetouch on something that looks like it, however. They all consider formal-ized plural languages, and in these they define a plural “identity” predicate≈, basically as an abbreviation.14

The effect (achieved in slightly different ways by different authors) isto 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-

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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.15

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’.

2 ‘is one of’ does not express a 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

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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 it does, we have a good argument for NTI. If itdoes not, we do not violate the metaphysical thesis that identicals may be“substituted”, and we may reject the argument for NTI from Leibniz’ Law.We now turn to an account of ‘is one of’ on which it does not express arelation.

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.

Consider what happens when we say

One of John and Paul is a musician.

Is the effect of this to predicate a property of John and Paul? We suggestthat it is not; instead, it is to distribute the property of being a musician,albeit in a different way than in the example with ‘each of’.

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One of is a musician.

does not express a property. Rather, when the blank is filled in, the resultexpresses a distributed proposition involving only the property of beinga musician. The property is disjunctively or existentially distributed withthe effect that the result is true just in case one of John and Paul has theproperty.

Reconsider

John is one of .

I take it that this is equivalent with

One 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.

If this is right, then ‘is one of’ does not function by expressing a rela-tion.

One way to see the semantic role of ‘is one of’ is through its contribu-tion to the truth-conditions of a sentence that contains it. We will give amore detailed discussion of the semantics of plural language in section 4,but it will help to look at one element of our theory now. We take a pluralterm (except in the limit case in which it is semantically like a singularterm) to refer to (each of) many things. A singular term refers to exactlyone thing. We suggest that the truth-conditions for something of the form

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:there is something t refers to and which is identical with somethingthat s refers to.

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

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

These truth-conditions ascribe a “meaning” (in a thin sense, at least) to‘one of’ without appealing to a relation.

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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 propertyover 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 susceptible toat least two readings, then there would be no unambiguous sentence thatbegins ‘John and Paul. . . ’, for the ambiguity of this phrase would neverbe resolved. Similarly, if you thought that every occurrence of a pluralpredicative expression is ambiguous, there would again be no unambigu-ous sentence that begins ‘John and Paul. . . ’, for whatever follows the dotswould be ambiguous. Thus, we suggest, the ambiguity of the sentence isnot due to there being more than one way to read ‘John and Paul’, nor is

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it due to there being more than one way to read ‘are heavier than Ringo’.‘John and Paul’ is semantically univocal, as is ‘are heavier than Ringo’.(Or, the latter may as well be, for our purposes; we may ignore any sensesof ‘heavier than’ other than the central one, if there are any.)

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:

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

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,

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‘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

(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 : À

is equivalent with

NOT ( non-F :: each of : À )

In English, we have the equivalence of the forms

One of À is . . .

and

It is not the case that each of À 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.

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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

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’.

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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.)

A application of NTI within the theory

These are the basic elements of our account: ‘each of’, ‘one of’, and ‘collec-tively’ serve to indicate three different modes of plural predication. Noneof them expresses a relation that might hold between some things andsome item (whether an object or a property). The result of adding, to anyof these modes, an expression that does express a property or relation (e.g.the ‘is’ of identity to yield ‘is one of’) is not itself an expression that func-tions by expressing a (plural) property or relation. Rather, the semanticsof the resulting complex expression (e.g., ‘is one of’) includes two very dif-ferent logical elements: a mode of predication, and a property or relation(to be predicated in that mode).

Now, the denial of NTI is compatible with this account of the seman-tic role of ‘is one of’; we have opened up the possibility that NTI fails.But it does not follow from our account that NTI fails. Suppose that somethingsxx are (collectively) identical with (collectively) John and Paul. Sup-pose we distributively predicate (disjunctively or conjunctively) some prop-erty F “over those thingsxx”. Is that the same thing as distributively predi-cating F “over John and Paul”?16This is almost exactly the issue of whetherNTI is correct.

Using our notation, we may capture the issue with the question whetherthe following semi-formal expressions indicate the same proposition:

F :: one of : John, Paul.F :: one of : theyxx.

That will be so, we suggest, if and only if: John and Paul, on the one hand,and theyxx, on the other, are “distributively identical”—i.e., if and only ifevery one of John and Paul is identical with one of themxx, and every oneof themxx is identical with one of John and Paul. For the two propositions

DRAFT


indicated by the semi-formal expressions distribute F “over some things”,producing a proposition that is true just in case one of them is F.

What about collective predication? Is the result of collectively predicat-ing F “of John and Paul” the same as the result of doing so “of themxx”?It will be if and only if the following semi-formal expressions indicate thesame propositions:

F :: collectively : John, Paul.F :: collectively : theyxx.

That will be so, we suggest, if and only if: John and Paul, on the one hand,and theyxx, on the other, are “collectively identical”— if John and Paul(collectively) are identical with themxx (collectively). Suppose so. Ouranalysis leaves it open whether this entails that the distributive predica-tions express the same propositions—NTI lets us close the gap, and an-swer yes.

We may reject NTI, and say that the gap remains open. But this raisesa question. If the semi-formal expressions

F :: one of : John, Paul.F :: one of : theyxx.

express different propositions, the difference is traceable, given composi-tionality, to a difference in semantic value of what comes after the singlecolon. If there is a difference there, how can

F :: collectively : John, Paul.F :: collectively : theyxx.

express the same proposition?We do not deny compositionality. The relevant expressions may not be

semantically identical. The latter two semi-formal sentences can expressthe same proposition because of what is done with the semantic valuesof the relevant expressions, as indicated by the expression for the modeof predication, ‘collectively’. Basically, the things ‘John and Paul’ refers to(or the things that ‘theyxx’ refers to) are (collectively) loaded into the blankspot of the property F. The results of these operations may be the same,even if ‘theyxx’ and ‘John and Paul’ are not semantically identical. Theresults will be the same if the things, the things that ‘John and Paul’ refersto, collectively are identical with the things that ‘theyxx’ refers to. ‘John

DRAFT


and Paul’ refers to each of John and Paul. ‘theyxx’ refers to each of themxx.So the two collective predications will be the same just in case John andPaul (collectively) are identical with themxx (collectively).

Objection: If those two expressions are not semantically identical, howcould John and Paul be identical, even “collectively identical” with themxx?Wouldn’t we have a semantic difference without an ontological difference?Is this some form of descriptivism?

Reply: Just because one of the things that ‘John and Paul’ refers to isnot identical with any thing that ‘theyxx’ refers to, it does not follow thatthe things that ‘John and Paul’ refers to are not (collectively) identical with(collectively) the things that theyxx refers to. (Note well that ‘John andPaul’ refers to each of John and Paul, not to John and Paul collectively;thus, the occurrences of ‘collectively’ in the previous sentence must becarefully placed and interpreted.) Of course, that is exactly what wouldfollow if NTI were correct. One is trying to reason from

Something is one of the things ‘John and Paul’ refers to, and is notone of the things ‘theyxx’ refers to

to

the things ‘John and Paul’ refers to are not (collectively) identical with(collectively) the things ‘theyxx’ refers to

which inference is a contrapositive application of NTI.

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

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À (collectively) are identical with Á (collectively)∴ φ(À) if and only if φ(Á)

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.

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.

DRAFT


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.17 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.18

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)

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︸︷︷︸ )

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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 singlethings.19 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.20

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 verbal

DRAFT


code 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.21

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.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:

DRAFT


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

Sometimes

À (collectively) are identical with Á (collectively)∴ φ(À) if and only if φ(Á)

is invalid—when φ does not express a property. ‘is one of’ can create sucha context; what others?

Our answer is: contexts analyzable as ones in which the “blank spot”is the operand for the mode-of-predication operators ‘one of’ or ‘each of’.

Some examples will convey the idea:

À are bandmates

is analyzable as

there is a band that À each are in

À admire one another

is analyzable as

each of À admires only other ones of À

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

DRAFT


À conversed

may be analyzable as

there was a conversation, and each of À 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.22

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.)

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.

DRAFT


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.

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:

DRAFT


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.23 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

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))

DRAFT


(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 everyproperty (collectively) possessed by (or relation collectively born by) thesexxis also possessed by thoseyy, and yet does not entail that it not be the casethat there is something that is one of thesexx and not one of thoseyy. ‘is oneof’ does not express a relation that holds between a thing and some things(collectively). There is a reasonable semantical theory for plural languagesthat 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

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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

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

DRAFT


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.

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.24 And since it is identity, itis a one-one function: for any thingsxx and any thingsyy, and any thing

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x, 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 “distrib-utively 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,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).

DRAFT


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.25

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 contradictionensues.26

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.27 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

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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.28 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. Thisrelation holds between a cat and its tail, or between a cat and a watermolecule that it has ingested, but not between a cat and a steel ball thatit has swallowed, nor between a cat and some shrapnel lodged in it. Thisis not the place to rehearse and expand on such reasons in detail; thereare well-known temporal and modal considerations—living things growand diminish, and could have been made of different parts. An under-appreciated reason, I believe, is this: there is no explanation of the fact thatthe water molecule is i-part of the i-fusion of the cat and the water mole-cule. But there is an explanation (in terms of life-processes) of why the

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water molecule counts as part of the cat (and why, for example, a steel ballthat it has swallowed does not). If this or other such arguments are correct,the notion of i-part, the one relevant to collective predication, is a differentnotion from the notion of organic part.

5.4 Singularities and RNTI

RNTI tells us that NTI holds for a restricted “range of things”: it holds ofsingularities. Thus, if our plural (and singular) quantifiers were restrictedin such a way as to range only over singularities, then NTI would holdafter all—for the restricted language. Would we be missing any expressivepower? Perhaps not, if all things are (identical with some) singularities. If so,we will show, then it is possible to give a semantics for a language in whichNTI fails, in a meta-language in which NTI holds.

Suppose that

Everything is identical with (collectively) some singularities—that is: for any thing x, there are some singularities xx such thatxx are (collectively) identical with x.

Then for any thing, there are some singularities that are (collectively) it.We can exploit this fact, given RNTI, to give a semantics for a plural

language, using a meta-language in which we quantify only over singu-larities. If we restrict the quantifiers of our meta-language to singularities,we can still say everything we could say without the restriction, with alittle ingenuity. The effect of this restriction is that the sentences

For any thing, it is a singularity.For any things, each of them is a singularity.

become true in our meta-language. Where, before, we had a thing iden-tical with some singularities (collectively), we now just have those singu-larities. But any property had by the old thing is had (collectively) by the(new) singularities (since the old thing was identical with them). So, es-sentially, nothing has been lost; we can still effectively refer to the thing,by referring to the singularities that are (collectively) it. (Given the uncon-troversial RNTI), in our restricted language, Material NTI holds.

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The technique applied to an example

Consider again the example in which three notes are played on the piano,supposing that each note is a singularity. Consider the CG dyad, x. Where,before, we would have said, for example

x satisfies ‘is a harmonic fifth’

we replace this with

c and g (collectively) satisfy ‘is a harmonic fifth’

Similarly, in our semantic account, we will not say that

the CG dyad

refers to x (the CG dyad). We will say that it refers to (collectively) c and g.In general, where we would have said a singular term refers to something,we now say it refers to (collectively) some singularities (sotto voce: the onesthat i-compose that thing). We will say that predication of a predicateF of a singular term t results in truth (or satisfaction) when the things(collectively) referred to by t (collectively) satisfy F.

Things get trickier when we consider plural terms that refer to morethan one non-singularity. Where, before, we would have said that

Each of x and y satisfies ‘is a consonant interval’

we now must say

c and g (collectively) satisfy ‘is a consonant interval’; andc and e (collectively) satisfy ‘is a consonant interval’.

Where, before, we might have said

x and y (collectively) satisfy ‘is a major triad’

we now must say

c, e, and g (collectively) satisfy ‘is a major triad’.

Similarly, in our semantic account, we will not say that

the CG dyad and the CE dyad

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refers to x and refers to y. Instead, we will say that it refers to c and g(collectively) and refers to c and e (collectively). It does not refer to c, norto e, nor to any singularity; nor does it refer to e and g (collectively), nor toc, e, and g (collectively).

When we interpret

each of the CG dyad and the CE dyad is a consonant interval

we calculate it as true via this calculation: for any singularities that ‘the CGdyad and the CE dyad’ refers to (collectively), they (collectively) satisfy ‘isa consonant interval’.

But when we interpret

the CG dyad and the CE dyad (collectively) are a major triad

we calculate it as true because: c, e, and g (collectively) satisfy ‘is/are amajor triad’. But how did we get those three out of the reference of theplural term?

The key to interpreting collective predication

The general principle in effect here is articulated, as applied to ‘the CGdyad and the CE dyad’:

The singularities xx such that: (1) every one of xx is one of somesingularities that (collectively) ‘the CG dyad and the CE dyad’refers to; and (2) any singularity that is one of some singulari-ties that (collectively) ‘the CG dyad and the CE dyad’ refers tois one of xx—those singularities, xx, collectively satisfy ‘is/area major triad’.

Note that we cannot simplify this. The singularities xx cannot be called“the singularities that the plural term refers to”. The plural term does notrefer to any one of c, e, and g! That definite description is infelicitous (andempty).29

The trickiest part of all involves the interpretation of plural quantifiers.We need an appropriate notion of the semantic value(s) of a plural variablerelative to an assignment. We think of the assignment as a relation, thatrelates variables to things in a very flexible way: it can relate the variableto a single singularity, or it can relate the variable to (collectively) many

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singularities; and it can do this more than once. For example, it may relatea variable to (collectively) singularities α and β, and also to (collectively)singularities β and γ, without relating the variable to α or to (collectively)α and γ.

One-place predicates will get a similar kind of interpretation, but withthe possibility of emptiness. n-place predicates require n + 1-place “refer-ence” relations.

You cannot get something from nothing. The expressive power of pluralquantifiers in the object-language is captured only by the use of these as-signment relations in the meta-language. Accordingly, we have to makeserious, though plausible, assumptions, in the meta-language, about theexistence of relations. We do not have the room to pursue the details here.

The reduction of non-singularities

We have seen that if everything is identical with some singularities, thenwe can give a semantics in which we only quantify over singularities.

This fact can be taken in different ways. It might be taken as an inter-esting “theorem” that follows from the logic of plurals and identity. Butit might be taken to show that the object-language, in which Material NTIfails, ought to be thought of as some kind of code for the meta-language, inwhich Material NTI holds. Further, it might be said that NTI is actuallycorrect, for a certain kind of language, (e.g., our meta-language) and that, inprinciple, such a language exists, and that such languages are superior:they give a more transparent representation of the world. In a transpar-ent language, singular terms refer to metaphysically singular things, andplural terms refer to a unique number of these things. Languages in whichMaterial NTI fails are misleading, since, for example, a singular term mightrefer once, without referring to a metaphysically singular thing, if it refersto several singularities (collectively); from the fact that is a singular term,one cannot tell how many singularities it refers to. There is a kind of “nu-merical opacity” in their terms and quantifiers. The number of singular-ities a singular term refers to might be greater than one, and the numberof singularities that a plural term refers to cannot be determined by thenumber of singular-term identities connected to it by ‘is one of’.30 Again,it might be urged that a proper understanding of a numerically opaque lan-guage translates it into a numerically transparent language—one in whichNTI holds.

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This is perhaps the attitude that the compositional nihilist takes. Inthe best language, our terms and quantifiers range only over mereologi-cal atoms (singularities). In this language, it is true to say “there are nocomposite objects”. Such a language exists, in principle. But ordinary andeven scientific language is not like this. In it, Classical Atomistic Mereol-ogy (CAM) is true, and there are lots of composite objects. In fact, CAM islogically valid, given the semantics or interpretive technique that we havesuggested. That technique, however, allows us to systematically translatesuch a language into the “superior” language in which NTI holds, and inwhich there is not even apparent quantification over anything but atoms.

Those who would take the line that NTI is correct might give a similarresponse to my argument against NTI. They could say to me

All along, we were only talking about good plural languages,and these are what you call “numerically transparent”. In them,NTI holds, and ‘is one of’ effectively expresses a relation—what you would call the “i-part” relation. What you have shownis, in effect, how to construct a bad plural language—one inwhich something that looks like Material NTI fails—out of agood one. NTI is valid, and your languages in which NTI ap-pears to fail should be (and can be, by the interpretive tech-nique of this section) regarded as (misleading) code for a lan-guage in which NTI holds.

They might go on to say that maybe English is numerically opaque; but wealways knew that ordinary language is logically obscure. But they couldalso insist that English itself is numerically transparent.

Perhaps they are right (however it goes with English). One reason Ido not agree is that I do not see what justifies the assumption that every-thing is identical with some singularities. If there are any non-singularities,then, for them, composition is identity. What would then justify this as-sumption, tantamount to mereological atomism, for these things? Theredoes not seem to be a compelling a priori argument for this kind of atom-ism, nor is there compelling empirical evidence for it. But my opponent’sprogram simply fails if this assumption is not right. Even if it is right, isit an axiom of metaphysics or of logic? If the former, NTI is not logicallyvalid. Further, even if NTI is valid in some “perfect” language, it is notat all clear that an “imperfect” language has no logic; if NTI is invalid in

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some language, then it does not have the status of simply being logicallyvalid. NTI is far from certain.

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Notes

1Acknowledgements: This paper has benefited from conversations between its au-thor and Mark Bedau, Ben Caplan, Eddie Cushman, Mark Hinchliff, David Kaplan, D.Anthony Martin, and Rotem Rabinovich. I am especially indebted to Gabriel Uzquiano,for a stimulating discussion and a very penetrating question about some of my earlierwork on this issue.

2It 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.

3For the argument that this sentence has no first-order formalization, not even an in-direct one, see [3], especially p. 433.

4Boolos argued that plural quantificational phrases ought to be taken seriously, de-spite their being strictly more powerful than the singular quantifiers of first-order logic,and that the logical role they play makes them suitable for interpreting monadic second-order quantifiers. See [3] and [4]. He also suggested that, on the side of formal semantics,the plural variables, bound by plural quantifiers, be interpreted in a way that simply gen-eralizes the interpretation of singular variables: roughly, a plural variable can have manyvalues; a variable “refers” (relative to an assignment) possibly more than once. On p. 336of [4], Boolos gives a formal semantics for the plural variables which uses “second-ordervariables”, in the meta-language, with the idea that the meta-linguistic variable be inter-preted as itself a plural variable. He is willing to gloss his semantics, on p. 337, as oneon which second-order (plural) variables have many values. He is not wedded to thisgloss, though it fits the formal idea very well. Thus a single valuation may relate a pluralvariable to an object, and also to another object—valuation is a relation, not a function.

5See, e.g., the works by Yi cited in this paper.

6Versions of this thesis have been defended by Donald Baxter, in [2] and [1], and byDavid Lewis, in [7]. Baxter takes the view very seriously, while Lewis holds a very weakversion of it. On Baxter’s view, whether x = y is relative to a something he calls a count.(Roughly, on one way of counting, x and y count as the same thing, while on another,they do not.) Lewis’ version is so weak that Lewis himself makes clear that it says onlythat composition is analogous to identity, and Lewis explicitly distances himself from aposition like Baxter’s. The view we are entertaining is different from either: compositionliterally is no different from identity—it’s not just analogous. And it is not relative, either.

7See Lewis [7] and Sider [12] for discussion of the virtues of composition as identity.

8Ted Sider, in [12], considers how general properties of the part/whole relation might

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be derived, if composition is identity. Sider argues against composition as identity, how-ever; his arguments turn on the use of NTI. The present paper arose, in part, from con-sideration of how the advocate of composition as identity might reply to Sider.

9This 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.

10The 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 [11].)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.)

11As of yet, there is no standard way of writing formal plural variables. Some alterna-tives include

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.

12We will sometimes harmlessly conflate ‘is identical with’ and ‘=’.

13One of the central arguments for Byeong-Uk Yi’s negative answer to the question “Ismereology ontologically innocent?” [18] turns on this issue. (The argument above is verymuch like one that Yi uses.) Yi crucially assumes, as I understand him, that NTI is correct.He rightly concludes, given NTI, that the claim that composition as identity, understoodin the most straightforward way, is logically incoherent.

14See section 4 of [17], chapter 6 of [8] and the Appendix of [10].

15He 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 in

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chapter 6 of [8], 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.

16The scare quotes are present because we are worried about possible ambiguity in thephrase, since no mode of predication has been included. This ambiguity should not beresolved by inserting ‘collectively’, however! “To disjunctively distribute F over (collec-tively) John and Paul” is, we take it, either nonsense, or just means either “to predicate Fof (collectively) John and Paul” or “to disjunctively distribute F of John and Paul”. Ourmodes of predication cannot be embedded meaningfully in each other.

17The term seems to come from Leonard and Goodman [6].

18As in [9] and [14].

19Of 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.

20See section 5 of [17].

21This 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.

22I believe that this observation will greatly help the “nihilist” to respond to GabrielUzquiano’s arguments in [16]. 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.

23It 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 plenitude

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of these reference relations.

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

24But only so to speak: as explained in section 3, I do not believe there is any point,metaphysically, in plurally plural quantification.

25Say 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.

26Gabriel Uzquiano makes essentially the same point, and further points about thedifficulties of combining unrestricted mereology, set theory, and plurals in [15]. Uzquianois skeptical that the two full theories can be satisfactorily combined in full generality.

27Here is one way to formulate CM, using plural quantification. (This formulationtraces back to Tarski [13].) 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.

28The least upper bound for xx is the thing x such that (1) for any y that is one of xx,y ≤ x (y is i-part of x) and (2) x is the least such thing, i.e., is i-part of any such thing.

29To make this perfectly clear, consider a more ordinary example of a similar issue,taken from McKay: suppose that in the library, there are some students meeting together,and there are some other students meeting together. (See Thomas McKay’s [8] for a care-ful discussion of the way ‘the’ behaves in conjunction with plural noun phrases.) Let itbe that students a, b, and c are meeting together in Room 101, and students d, e, and f aremeeting together in Room 102, and no other students are meeting together in the library.Then the phrase ‘the students who are meeting together in the library’ is infelicitous be-cause of a failure of uniqueness. Note that if we regard the description as picking out allsix students, we get the unhappy result that it is true that

The students who are meeting together in the library are not meeting to-

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gether in the library.

It’s better to regard the description as infelicitous. It’s better to regard the displayedsentence as semantically problematic (in this circumstance), and truth-value-less or false,rather than as straightforwardly true.

30In our semantics for a language in which NTI fails, there is still a unique numberassociated with a plural term: the number of “times” it refers. This shows up in theobject language as follows: if a plural term t refers finitely many times, there will be aunique n for which the sentence of the form

∃x1∃x2 . . . xn(OneOf(x1, t) ∧ OneOf(x2, t) ∧ . . . ∧ x1 6= x2 ∧ x1 6= x3 ∧ . . .)

is true. The reason one might call a language in which NTI fails “numerically opaque” isthat the unique number associated with a referring term is independent of the number ofsingularities referentially associated with it.

References

[1] Donald L. M. Baxter. Identity in the loose and popular sense. Mind,97:575–582, 1988.

[2] Donald L. M. Baxter. Many-one identity. Philosophical Papers, 17:193–216, 1988.

[3] George Boolos. To be is to be a value of a variable (or to be somevalues of some variables). Journal of Philosophy, 81(8):430–449, 1984.

[4] George Boolos. Nominalist platonism. Philosophical Review, 94(3):327–344, 1985.

[5] Peter Van Inwagen. Material Beings. Cornell University Press, 1990.

[6] Henry Leonard and Nelson Goodman. The calculus of individualsand its uses. Journal of Symbolic Logic, 5:45–55, 1940.

[7] David Lewis. Parts of Classes. Blackwell, 1991.

[8] Thomas McKay. Plural Predication. Forthcoming.

[9] Adam Morton. Complex individuals and multigrade relations. Nous,9:309–318, 1975.

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[10] Agustin Rayo. Beyond plurals. In Absolute Generality. Oxford Univer-sity Press. Forthcoming.

[11] Gideon Rosen and Cian Dorr. Composition as a fiction. In Richard M.Gale, editor, The Blackwell Guide to Metaphysics. Blackwell, 2003.

[12] Theodore Sider. Parthood. Forthcoming in a volume of Topics in Con-temporary Philosophy.

[13] Alfred Tarski. Foundations of the geometry of solids. In John Coco-ran, editor, Logic, Semantics, Meta-Mathematics. Hackett, 1983.

[14] Barry Taylor and A. P. Hazen. Flexibly structured predication. Logique& Analyse, 139–140:375–393, 1992.

[15] Gabriel Uzquiano-Cruz. Th price of universality. Philosophical Studies.Forthcoming.

[16] Gabriel Uzquiano-Cruz. Plurals and simples. The Monist, 87(3):429–451, July 2004.

[17] Byeong-Uk Yi. The logic and meaning of plurals II. Journal of Philo-sophical Logic. Forthcoming.

[18] Byeong-Uk Yi. Is mereology ontologically innocent? PhilosophicalStudies, 93(2), February 1999.

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