Identity and Plurals

Transcription

1 Identity and Plurals Paul Hovda February 6, 2006 Abstract We challenge a principle connecting identity with plural expressions, one that has been assumed or ignored in most recent philosophical discussions of the logic, semantics, and metaphysics of plurals. We call it the principle of the Numerical Transparency of Identity (NTI). It says that for any plural terms t and s, the sentence t are identical with s (interpreted collectively at both nouns, not distributively) is logically equivalent with the sentence Every one of t is one of s and every one of s is one of t. There are reasons one might want to deny this principle. For example, one might hold that composition is identity: when some things compose a thing, they are identical with it. But this is logically coherent only if NTI fails. We show that the denial of NTI is logically coherent; hence NTI is not a correct logical principle. The key steps of the case against NTI include an analysis of collective and distributive plural predication that motivates our claim that is one of does not express a relation; a picture of the metaphysics of plural properties and relations that motivates our view of collective plural identity predications; and a clear semantic theory for a language in which NTI fails, which meshes with the analysis and the metaphysical picture.

2 Contents 1 The Question The numerical transparency of identity Composition as identity and the failure of NTI NTI and contemporary formalisms of plurals is one of does not express a relation 9 3 Plurals, properties, and relations 19 4 Semantics 25 5 Glimpses Beyond Restricted NTI and mereological notions For any things, a thing Composition as identity Singularities and RNTI

3 Identity and Plurals 1 1 The Question 1 The aim of this paper is to give good arguments against a certain logical principle involving plural constructions, connecting distributive and collective predications of identity. 2 This principle seems to have been ignored, or tacitly assumed without careful consideration, in recent developments in the project of formalizing the language and logic of plurals. Its failure seems to go hand in hand with the idea that composition, or at least one kind of composition, is identity. The principle, which I call the Numerical Transparency of Identity (NTI) says that any construction of the form ➀ (collectively) are identical with ➁ (collectively) is logically equivalent with the corresponding instance of for all things, that thing is one of ➀ if and only if it is one of ➁ where the substituends for ➀ and ➁ are appropriate roughly, are referential 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 what count as appropriate substituends in more detail, momentarily. First, a little background. One of the limitations of standard first-order logic is that plural constructions, and reasoning involving them, are not directly formalizable in it. There are two logically important aspects of plural constructions that cannot be captured with the ideas of first-order logic: distributive and collective predications involving plural nouns. A famous example of a distributive use that cannot be straightforwardly given a first-order treatment is the Geach-Kaplan sentence: Some critics admire only one another. 3 This is distributive because of the phrase one another ; roughly, the admires relation is distributed (in a particular way) over the critics. An example of a collective use is John and Paul (together) lifted the piano. This is equivalent to no truth-function of John lifted the piano and Paul lifted the piano, and, hence, cannot be straightforwardly given a first-order treatment.

4 Identity and Plurals 2 For the most part, in the logical tradition following Frege, the limitation of first-order logic to singulars has not been seen as a problem: most things you might want to say with a plural construction can be re-cast in singular language. You have a surrogate for distributive uses if you can talk about sets, which are single things that can bear the membership relation to many things. Instead of distributing a property directly over many critics, you can say: there is a set such that every member of it is a critic who admires only other members of it. And you have a surrogate for collective uses if you can talk about mereological fusions, or some other kind of single things composed of many things. Instead of saying that John and Paul collectively lift something, you say that the fusion of John and Paul lifted something. The distributive use of John and Paul in this surrogate can then be eliminated with a set. So plural constructions (and thus the principles of reasoning involving them) have not been directly formalized, since they can be simulated, up to a point, with surrogate singular constructions. The only obvious catch is that you need a bigger ontology than the plurals seem to require. Thus, as first-order logic and set theory were developed, there were few attempts to give plural constructions a logical life of their own none that caught on. There is not yet a standard way to extend the formalization we call first-order logic to accomodate plurals directly. But that is changing; the last twenty years have seen a number of developments in this direction, accompanied with suspicion about the singular surrogates; the work of George Boolos was perhaps the primary spur. 4 Boolos work suggested that the distributive uses of plurals are logically powerful and yet ontologically innocent: one can have the expressive power of the surrogate (sets with many members), without commitment to single objects that encompass the many. Boolos made less use of the collective uses of plurals; the work of Byeong-Uk Yi, among others, has gone some way toward formalizing these uses as well. 5 Boolos and Yi both offer good criticisms of the use of surrogates. The topic of this paper is a principle that must be considered if we are to give a formalization for plural constructions in their own right. But I know of no explicit discussion of it; I believe I see it implicitly assumed in contemporary treatments, but it is often difficult to tell whether it is being assumed or ignored. It is difficult to grasp what NTI says, because we are not 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

5 Identity and Plurals 3 (collectively) are heavier than George. The sentence John and Paul (collectively) are identical with George seems to have exactly the same grammatical form. There are fairly obviously true collective identity sentences, like John and Paul (collectively) are identical with John and Paul (collectively) and John and Paul (collectively) are identical with Paul and John (collectively). These appear to have the same grammatical form as the readily intelligible John and Paul (collectively) are heavier than George and Ringo (collectively). But why might one think NTI holds or fails? Against it, I believe that the best way to understand the thesis that composition is identity is as the thesis 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 reasons. First, it yields a kind of ontological innocence, of composed objects: if the composed is literally identical with the composers, then ontological commitment to the composed is not a further commitment beyond commitment to the composers. 7 Commit yourself to some things, and you, thereby, entitle yourself to all the things composed of them. Second, if composition is identity, then, perhaps, the logical properties of the part/whole relation can be deduced from, or at least illuminated by, what we know about the logic of the identity relation. For example, the transitivity of parthood might be derived from the transitivity of identity. 8 There are problems for the view that composition is identity that have nothing to do with NTI, but which turn on the temporal and modal differences between a typical composed object and its composers. If a living being is composed of different parts at different times, then how could it be identical with the parts it is composed of at one time? If it could have been composed of different parts, how could it be identical with the parts it is actually composed of? These problems can be addressed in a metaphysical framework like David Lewis, with four-dimensionalism and counterpart theory. But even if we do not accept such a framework, and we reject composition as identity for living beings or other things, it may still be that there is a kind of composition (and an attendant kind of part/whole relation) that is identity. Here is an example that all might accept: When you play two notes simultaneously on a piano, middle C and the G a fifth above it, you play a harmonic interval, or dyad. (Think token not type.) There are at least three numerically distinct sounds: the C, the G, and the dyad. If you can hear

6 Identity and Plurals 4 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 other two sounds. But it is impossible to hear the dyad without hearing the two notes, 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 thing possible while the others are impossible? Answer: because the two notes are the dyad. More precisely, the C and the G (collectively) are identical with 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 kind of composition, motivates the rejection of NTI; 10 but there is a serious argument for NTI (and hence, against composition as identity) that must be addressed. NTI is derivable from a substitutivity priniciple governing plural expressions, reminiscent of the familiar substitutivity of identicals for singular expressions. Suppose we knew that collective identity supports substition for any context φ; that is, ➀ (collectively) are identical with ➁ (collectively) logically implies φ(➀) if and only if φ(➁) where φ(➁) arises from φ(➀) by substituting one or more occurrences of ➀ in φ(➀) with ➁. Then we would know that NTI is true: we simply let φ(➀) be the indisputable for all things, that thing is one of ➀ if and only if it is one of ➀ and substitute at the second occurrence of ➀. I will present three ideas about the logic and metaphysics of plural constructions that will reveal why this kind of substitution might fail. Once this argument from substitution is rejected, NTI should be in serious doubt. (For what else would motivate NTI?) First, I will argue that is one of does not function by expressing a relation. Thus, the failure of substitution here does not violate the law that identicals are indiscernible that identicals have the same properties and bear the same relations. The argument flows from an analysis of the central kinds of plural predication. Second, I will give a metaphysical picture of collective relation-bearing that shows how we should think of collective identity and the kind of indiscernibility it entails. On this picture, there

7 Identity and Plurals 5 is no fundamental distinction between plural and singular relations, and every relation has a fixed arity that does not vary, no matter how many objects are involved in a given bearing of it. For example, the relation to lift 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 some things (collectively) to some things (collectively). But if it does relate some things to some things like this, then any property had by the first things (collectively) will be had by the second things (collectively). Third, I will sketch a semantics for plural terms that will show how is one of functions without expressing a relation. In brief, a plural term t refers to each of some things: n is one of t (with singular term n) will be true just in case the object that n refers to is also referred to by t. The semantics, combined with 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 explicit about this. A singular term as a substituend for ➀ in the form ➀ (collectively) are identical with ➁ (collectively) results in an ungrammatical sentence of (American) English if we keep the words exactly as written. So, as a limit case, we will count the following as an instance of this form John is identical with John and Paul (collectively). When a singular term is a substituend for ➁, the collectively is not needed or wanted, so we will also count both of these: John and Paul (collectively) are identical with John. John is identical with John. as instances of this form. It is also convenient to stretch English plurals to include singulars in a couple of other ways. We will extend our conception of the form for all things, that thing is one of ➀ if and only if it is one of ➁

8 Identity and Plurals 6 similarly, except that for this, we do not need to change any words in the result of the substitution; we need only to ensure that the result of substituting singular terms (for either ➀ or ➁) is recognized as grammatical by the reader. Accordingly, in the rest of the document, I will assume that it is 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 things xx,... we mean to include, as a limit, that there is exactly one thing that is one of them xx. (Just as we may use subscripted singular first-order variables on singular English nouns to indicate anaphoric connections, we will use subscripted plural variables, as above, to indicate anaphoric connections among plural nouns.) 11 This usage may not be standard English, but it is generally followed in the contemporary study of the logic of plurals. Note that with this usage, the following is a logical truth: 12 For any thing x, there are things xx, such that for every thing y: y is one of them xx if and only if y = x. NTI is a principle saying that two forms are logically equivalent. Corresponding to it is the sentence that represents its universal generalization. Let us call this sentence, For any things xx and any things yy : they xx are (collectively) identical with them yy (collectively) if and only if for any thing, it is one of them xx if and only if it is one of them yy. 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 one that says that ➀ (collectively) are identical with ➁ (collectively) logically implies for all things, it is one of ➀ if and only if it is one of ➁

9 Identity and Plurals 7 It will be useful to have short labels for these forms. Let us say that the first proposes the collective identity of ➀ with ➁. Let us say that the second proposes the distributive identity of ➀ with ➁. I am suggesting that collective identity does not imply distributive identity. I see no reason to deny the other half of NTI: I accept that distributive identity implies collective identity. The principle is called the Numerical Transparency of Identity because it tells us, in effect, that the collective identity of n things with m things entails that n = m. Thus, for any things, there is a unique number 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 = m. In that case, the distinct numbers n and m are equally applicable to them. Thus, to know that they are n in number may be compatible with their being m in number, and one may or may not know the latter; thus, their being m in number (or not) is not transparent to one who knows that they are n in number, in the way it would be if NTI held. It will emerge, on our view, that there is a unique number associated with each plural referring expression, while there is no guarantee that this number uniquely fits the things, the purely worldly component of what is referred 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 the thesis that composition is identity. An example will illustrate the conflict. Suppose that all composition is identity, and that classical mereology is correct. Then, for any four things, there is a mereological fusion of them, that is identical with them. (Not identical with each of them, but identical with them collectively.) Let a, b, c, and d be four cats, and let z be their mereological fusion. Let the fusion of a and b be x, and the fusion of c and d be y. Then, a, b, c and d compose z, and also x and y compose z. Since composition is identity, a, b, c and d (collectively) are identical with x and y (collectively). This is a statement of collective identity. Now suppose that NTI is correct. Then we can infer distributive identity from this collective identity, so that each of x and y is one of a, b, c and d. But this is simply false. 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

10 Identity and Plurals 8 (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 each of the others; e.g., a would be b. So if NTI is correct, composition is not identity. We can see also that if NTI is incorrect, then it looks like identity is, or is one kind of, composition. Suppose NTI fails. Then there can be some things (call them the Xs ) (collectively) identical with some things (call them the Ys ) and some thing x that is one of the Xs and not one of the Ys. x is not one of the Ys, but it is intimately connected with the Ys, since it is one of some things that are identical with the Ys. One might say its identity overlaps the identities of the Ys, and is somehow entirely overlapped. The connection is perhaps clearer in the special case in which there is only one thing y that is one of the Ys; i.e., every thing that is one of the Ys is identical with 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 it in this discussion.) Then the Xs are (collectively) identical with y. Thus x, though it is not identical with y, is one of some things that are (collectively) (identical with) y. The relation between x and y is not identity, but it is importantly connected to identity. One might say the identity of x is not entirely separate from the identity of y, or even that the identity of x is part of the identity of y. One might plausibly suggest, in more familiar vocabulary, that this intimate connection between x and y is the relation of part 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 semantics of plurals, ones rich enough to cover both distributive and collective predication, NTI is not directly discussed. (I have in mind the discussions of Yi [17], McKay [8], and Rayo [10].) Not directly discussed. All three touch on something that looks like it, however. They all consider formalized 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) is to define the plural identity predicate in terms of is one of. Let us use x and y as formalized singular variables, and xx and yy as formalized plural variables, and OneOf as a formal symbol with the syntax of a two-

11 Identity and Plurals 9 place relation, the first place of which admits formal singular terms, the second place of which admits formal plural terms. Yi directly defines as xx yy 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 much larger framework). McKay s treatment is slightly different, but basically has the same effect. 15 It might thus appear that these authors take NTI and Material NTI to be true by definition. But this is not, I think, a correct description of their 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, suggest that Material NTI is itself an analytic truth, or at least some kind of logical 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 distributive identity might appear to be a special case of the substituting of identicals. But the substitutivity principle that would be invoked to yield NTI is a principle about the substitution of terms, not of things. We should not accept that principle as a proper principle about identity itself and things themselves. Speaking of things, we should accept that: for any things xx and any things yy, if these xx (collectively) are identical with those yy (collectively), then any property that these xx (collectively) have is a property that those yy

12 Identity and Plurals 10 (collectively) have. This is a proper, metaphysical, principle of the substitutivity 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 know that is one of expresses a relation. If it does, we have a good argument for NTI. If it does 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 a relation. 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 a musician of John and Paul? Or is it to predicate a property, the property of being a musician, in such a way as to yield a proposition that is true just in case each of John and Paul has the property? The latter is a reasonable view. If so, Each of is a musician. does not express a property. Rather, when the blank is filled in, the result expresses what me might call a distributed proposition involving only the property of being a musician. The property is conjunctively or universally distributed, with the effect that the result is true just in case each of John and 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 suggest that 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.

13 Identity and Plurals 11 One of is a musician. does not express a property. Rather, when the blank is filled in, the result expresses a distributed proposition involving only the property of being a musician. The property is disjunctively or existentially distributed with the effect that the result is true just in case one of John and Paul has the property. 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. It distributes the property of being identical with John, and it distributes it disjunctively. If this is right, then is one of does not function by expressing a relation. One way to see the semantic role of is one of is through its contribution to the truth-conditions of a sentence that contains it. We will give a more 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 plural term (except in the limit case in which it is semantically like a singular term) to refer to (each of) many things. A singular term refers to exactly one 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 refers to; that is: there is something t refers to and which is identical with something that 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.

14 Identity and Plurals 12 Modes of plural predication Let us reflect a little further on the idea that, roughly speaking, one of and each of serve to indicate a particular way of distributing a property over some things. We will suggest that all ways of connecting a property with some things must be connected by one and only one mode of connection, where these modes include the distributive modes and the collective mode. Consider the English sentence John and Paul are heavier than Ringo. This sentence is ambiguous. There are at least two readings, given, apparently unambiguously, by and John and Paul are each heavier than Ringo. John and Paul collectively are heavier than Ringo. But are these really unambiguous? Maybe they are each ambiguous in exactly the same way as the original. Consider the following further disambiguations and John and Paul collectively are each heavier than Ringo. 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 if you thought that every occurrence of a plural noun phrase is susceptible to at least two readings, then there would be no unambiguous sentence that begins John and Paul..., for the ambiguity of this phrase would never be resolved. Similarly, if you thought that every occurrence of a plural predicative expression is ambiguous, there would again be no unambiguous sentence that begins John and Paul..., for whatever follows the dots would be ambiguous. Thus, we suggest, the ambiguity of the sentence is not due to there being more than one way to read John and Paul, nor is

15 Identity and Plurals 13 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 senses of 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 made between (the phrases) and John and Paul 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 (if not in English). F might be disjunctively distributed, so that the resulting proposition is logically equivalent with John is F or Paul is F. F might be conjunctively distributed, so that the resulting proposition is logically equivalent with John is F and Paul is F. Finally, F might be collectively predicated, so that the resulting proposition is logically independent of any truth-function of and John is F 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, presented in the same order:

16 Identity and Plurals 14 F :: one of : John, Paul. F :: each of : John, Paul. F :: collectively : John, Paul. The semi-formal schemes are unambiguous representations of the three propositions 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 meaningfully embedded or iterated. Further, John and Paul each is not a nounphrase on a par with John and Paul, to be treated semantically simply in terms of its reference. Similarly for John and Paul collectively. The semantic difference between John and Paul each and John and Paul collectively is not to be thought of as like that of two noun phrases that refer differently. The only referring is done by John and Paul ; the more complex noun phrases represent the reference together with a logical operation. Yet, for John and Paul to be used as a subject for predication, some mode of predication must be attached. You cannot simply refer plurally and then predicate a property. Again, John and Paul are heavier than Ringo is ambiguous not because John and Paul is ambiguous, nor because heavier than Ringo is ambiguous; there is no ambiguity about what property is being predicated. The ambiguity of the sentence is due to under-specificity about the intended connection between the property and the objects referred to. Thus we should never ask ourselves whether some things have a given property or bear a given relation without having in mind a particular mode 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, in a sense, predicated of or, better, asserted of some things, without ambiguity but this operation includes two distinct logical components: the property to be predicated, and the mode of predication. These two components do not fuse to form another property. If they did, it would be ambiguous in which mode the resulting property was to be predicated. (And there would then seem to be no reason for the infelicitous further disambiguations not to make sense.) Thus, our theory about the English is one of is this: it does not express a relation, but is a combination of the is of identity with an expression,

17 Identity and Plurals 15 one of, that indicates what we have called a mode of predication, disjunctive distribution. Our semi-formal representation of John is one of John and 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 relation, to observe that there is a systematic equivalence between sentences involving each of and sentences involving one of, very much like that between and. Using our semi-formalism, for any property F F :: one of : ➀ is equivalent with NOT ( non-f :: each of : ➀ ) In English, we have the equivalence of the forms and One of ➀ is... It is not the case that each of ➀ is not... (Actually, this form is ambiguous: roughly, there is a scope ambiguity for the 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 relation between John, on the one hand, and John and Paul on the other, is equivalent 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.

18 Identity and Plurals 16 But surely each of, as used in this last sentence, does not function by expressing a relation; this is further evidence, we suggest, that is one of, in the original, does not either. One could hold, I suppose, that is non-identical with each of expresses a relation, and that this last sentence expresses the negation of the holding of this relation between John, on the one 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 holding of this relation between John, on the one hand, and John and Paul on the 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, for any 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 relations with special logical connections. It is much more plausible to separate the roles 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 mode of 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 the case, then we would have an excellent explanation of the logical equivalence of these sentences, and for a host of other logical equivalences. Then we would make a parallel move for one of.

19 Identity and Plurals 17 I am not sure, however, that the sentences of this pair express exactly the same proposition. But I am sure that I can give a picture of the semantics for (formal correlates of) each of and one of that makes clear why the sentences are logically equivalent, without the postulation of logical connections among properties. (I will fill out that sketch in section 4 after the discussion of the metaphysics of properties and collective property possession.) A application of NTI within the theory These are the basic elements of our account: each of, one of, and collectively serve to indicate three different modes of plural predication. None of them expresses a relation that might hold between some things and some item (whether an object or a property). The result of adding, to any of 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 functions by expressing a (plural) property or relation. Rather, the semantics of the resulting complex expression (e.g., is one of ) includes two very different 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 semantic 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 some things xx are (collectively) identical with (collectively) John and Paul. Suppose we distributively predicate (disjunctively or conjunctively) some property F over those things xx. Is that the same thing as distributively predicating F over John and Paul? 16 This is almost exactly the issue of whether NTI is correct. Using our notation, we may capture the issue with the question whether the following semi-formal expressions indicate the same proposition: F :: one of : John, Paul. F :: one of : they xx. That will be so, we suggest, if and only if: John and Paul, on the one hand, and they xx, on the other, are distributively identical i.e., if and only if every one of John and Paul is identical with one of them xx, and every one of them xx is identical with one of John and Paul. For the two propositions

20 Identity and Plurals 18 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 predicating F of John and Paul the same as the result of doing so of them xx? It will be if and only if the following semi-formal expressions indicate the same propositions: F :: collectively : John, Paul. F :: collectively : they xx. That will be so, we suggest, if and only if: John and Paul, on the one hand, and they xx, on the other, are collectively identical if John and Paul (collectively) are identical with them xx (collectively). Suppose so. Our analysis leaves it open whether this entails that the distributive predications express the same propositions NTI lets us close the gap, and answer yes. We may reject NTI, and say that the gap remains open. But this raises a question. If the semi-formal expressions F :: one of : John, Paul. F :: one of : they xx. express different propositions, the difference is traceable, given compositionality, to a difference in semantic value of what comes after the single colon. If there is a difference there, how can F :: collectively : John, Paul. F :: collectively : they xx. express the same proposition? We do not deny compositionality. The relevant expressions may not be semantically identical. The latter two semi-formal sentences can express the same proposition because of what is done with the semantic values of the relevant expressions, as indicated by the expression for the mode of predication, collectively. Basically, the things John and Paul refers to (or the things that they xx refers to) are (collectively) loaded into the blank spot of the property F. The results of these operations may be the same, even if they xx and John and Paul are not semantically identical. The results will be the same if the things, the things that John and Paul refers to, collectively are identical with the things that they xx refers to. John

21 Identity and Plurals 19 and Paul refers to each of John and Paul. they xx refers to each of them xx. So the two collective predications will be the same just in case John and Paul (collectively) are identical with them xx (collectively). Objection: If those two expressions are not semantically identical, how could John and Paul be identical, even collectively identical with them xx? 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 is not identical with any thing that they xx refers to, it does not follow that the things that John and Paul refers to are not (collectively) identical with (collectively) the things that they xx refers to. (Note well that John and Paul refers to each of John and Paul, not to John and Paul collectively; thus, the occurrences of collectively in the previous sentence must be carefully placed and interpreted.) Of course, that is exactly what would follow if NTI were correct. One is trying to reason from to Something is one of the things John and Paul refers to, and is not one of the things they xx refers to the things John and Paul refers to are not (collectively) identical with (collectively) the things they xx 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 properties, 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. We found good grounds to question it: is one of is a composite of identity and one of, and the latter (along with its dual each of ) expresses a mode of predication, and is not a relational expression. We now motivate a picture of properties and relations and the metaphysics of collective property-bearing, and the mode of plural predication we called collective, that will help us to see what the appropriate plural form of Leibniz Law really amounts to. This in turn will allow us to answer the more general question: for what kind of context φ is the form

22 Identity and Plurals 20 valid? ➀ (collectively) are identical with ➁ (collectively) φ(➀) if and only if φ(➁) Fixed arities for the multigrade It is crucial to our conception that, in short, all properties and relations have fixed arities, despite being able to admit more than one thing (simultaneously) in an argument place when predicated collectively of many things. We take for granted that there are objects, properties, and relations that somehow come together to form (atomic) facts and propositions, and that complex facts and propositions are somehow formed out of these. We take for granted that it is straightforward to see a systematic correlation between these entities and the sentences of a first-order language. Now suppose 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 the mold of any sentence in a first-order language. Whereas the (conjunctively) distributive reading of the plural predication reasonably can be thought 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-place relation, 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 say something like There are some men, and there are some relations, and each of those relations is a lifting relation, and those men bear one of those relations to the piano.

23 Identity and Plurals 21 Besides the implausibility of the suggestion of the covert quantification over relations, this approach faces the problem of clarifying the notion of a lifting relation, which would appear to be a new category of property of relations. Perhaps lifts expresses a single multigrade relation. 17 A multigrade relation can apply as if it had any number of blank spots. But what exactly does this mean? Most attempts to make this out have focussed on formal logic 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 John and Paul (collectively, not each), on the one hand, and the piano, on the other. The lifting relation involved is the very same two-place relation that 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 one of its places. The collective mode of predication is what connects (one of the blank spots of) the relation with John and Paul. This conception has a great advantage over conceptions on which what is really going on in this case involves a three-place relation (or three-place instance or determinate of a multigrade relation). Consider the difference between (the collective readings of) and John and Paul fight George and Ringo. John and Paul and George fight Ringo. If fight in both examples acts as a four-place relation, it would seem that the very same proposition, one we might represent as Fight(john, paul, george, ringo) is being expressed by both sentences. Clearly, the two English sentences are not logically equivalent. Thus it is much better to think of fight as once and for all expressing a two-place relation. We would then represent the propositions expressed by the two sentences as something more like ) Fight( john, paul }{{}, george, ringo }{{}

24 Identity and Plurals 22 and ) Fight( john, paul, george }{{}, ringo }{{} respectively. This conception generalizes neatly: we may think of every property and relation has having a fixed finite arity, but as accepting any thing or 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) in the one blank spot of any property; just to say that you get an (atomic, objectual) proposition, a logically basic possibility of a fact, directly involving some objects at any rate, you get something of the same kind as 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 operations like conjunction or quantification; nothing beyond predication, or predication in a mode. An atomic proposition involves merely a property, or relation, and proposed bearers of it. By an objectual proposition I mean roughly what is usually meant by a singular proposition a proposition that directly involves objects; for obvious reasons, that term is potentially misleading here.) One upshot of our conception is that there is no fundamental metaphysical distinction between plural properties and singular ones. There may be, of course, properties that are actually possessed only by single things. 19 But the propositions formed by predicating such a property exist nonetheless the point is that they are false. Similarly, there may be properties that are actually possessed, but never by single things. Again, the propositions formed by predicating such a property of one thing will simply be false, maybe necessarily false or even logically false, not nonexistent. We thus agree with Byeong-Uk Yi that some properties can accept many things as such (that is, collectively) as arguments, and generally that some blank spots in relations can accept many things (collectively) as arguments. Unlike Yi, we think that this is so for all properties and relations. 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

25 Identity and Plurals 23 code for plural talk, exactly because what it is for two thingses to have a property is either (1) for each of them to have it (in which case we have nothing new) or (2) for certain things collectively to have it; roughly these are the things you get when you put the two thingses together. To be exact, call these things things +. Things + are: the things such that (A) every one of them is (either one of one of the first thingses or one of one of the other thingses) and (B) every thing that is (one of one of the one thingses or 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 simultaneously 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 relation and 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 it will 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 identity relation as about substitution. Consider the position that says that the pseudo-relation we called distributive identity is in fact the identity relation, and that the plural relation we call identity is something short of identity call it sub-identity. The advocate of this position might say: We reply: Distributive identity guarantees substitution salva veritate in all contexts; sub-identity does not. Therefore, distributive identity deserves to be thought of as the plural identity relation, while sub-identity is a mere equivalence relation. If, as you say, distributive identity is not actually a relation, then there is no plural identity relation.

26 Identity and Plurals 24 Identity should be thought of first in terms of metaphysics, not language. Given our way of thinking of the multigrade, we may ask: Is there some relation such that of necessity: when it holds, of some things xx and some things yy, then the result of putting xx into the blank spot of any property, and the result of putting yy into it, will have the same truth value? If so, that is what deserves to be thought of as the identity relation. Sub-identity is such a relation, and may as well be the only such relation. There is no relation of distributive identity. There may be something that looks grammatically like a relation-symbol, but it is a potentially misleading way of expressing something analyzable in terms of each of, and hence, not a mere relation. Failures of substitutivity Sometimes ➀ (collectively) are identical with ➁ (collectively) φ(➀) if and only if φ(➁) is invalid when φ does not express a property. is one of can create such a 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: ➀ are bandmates is analyzable as there is a band that ➀ each are in ➀ admire one another is analyzable as each of ➀ admires only other ones of ➀ Sometimes verbs seem to involve implicit quantification over events. In these cases, we may have contexts like this. E.g.,

27 Identity and Plurals 25 ➀ conversed may be analyzable as there was a conversation, and each of ➀ partook in it Thus we hold that many English (plural) predicates do not simply express properties, but, rather, express operations involving one of and each along with some properties. Many apparent failures of substitutivity of identicals (in the plural) arise from this fact Semantics We have seen that it is a plausible claim that is one of does not express a relation, and that there is a reasonable picture of the metaphysics of collective property bearing that meshes with that claim. Building on these ideas, we now give the outlines of semantical theories for languages with plurals, 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 it does not simply express a property or relation). We do not assume Material NTI in our meta-language. We will be talking about the reference of singular and of plural terms in our objectlanguage, and we will collapse into this notion the idea of reference for variables, relative to an assignment. The crux of the theory is that singular terms refer only once, and plural 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 if and 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 the loose sense that there be some things, more than one, such that it refers to them (collectively). (This makes perfect sense on the assumption that NTI fails.) A plural term t may refer to more than one thing, in the strong sense that 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 to how many things it may refer to.