Compounds and Aggregates

Compounds and Aggregates Kit Fine Noûs, Vol. 28, No. 2. (Jun., 1994), pp. 137-158. Stable URL: http://links.jstor.org/sici?sici=0029-4624%28199406%2928%3A2%3C137%3ACAA%3E2.0.CO%3B2-C Noûs is currently published by Blackwell Publishing.

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http://www.jstor.org Fri Dec 14 13:40:57 2007

Compounds and Aggregates KIT FINE

University of California, Los Angeles

Some objects appear to be composed of parts: a quantity of sand of its grains, a throbbing pain of its throbs, a set of its members, and a proposition of its constituents. There seem to be two fundamentally different ways in which an object can be composed of parts. One is nonstructural in character; the parts just merge. The other is structural; the parts hang together within a structure. Thus of the examples above, the first two, the sand and the pain, are composed from their parts in a nonstructural fashion, while the last two, the set and the proposition, are composed in a structural manner. The notion of a nonstructural method of composition may be taken to be one which conforms to certain structure-obliterating identity conditions. These are as follows: order and repetition among the composing objects is irrelevant to the result; the composition of a single object is the object itself; and the composition of compositions of objects is the composition of those very objects1. Thus the first of these conditions excludes concatenation as a nonstructural method of composition; while each of the remaining conditions excludes the set-builder (the operation which composes a set from its members). Let us agree to call any nonstructural method of composition a method of fusion. There is a particular such method, I call it aggregation, which has been very prominent in the literature on part-whole. It may be characterized as a method of composition which conforms to the identity conditions above and which also conforms to the following existence conditions: the aggregate of objects which exist in time exists at exactly those times at which one of the objects exists; and an aggregate of objects which are located in space occupies, at any given time at which it exists, exactly those places which are occupied by one of the objects. It has often been supposed that aggregation is a legitimate method of composition, that objects may be composed from others in conformity with the conditions

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set forth above. What has made aggregation so attractive, apart from any intuitive appeal it may have, are two main factors (which will be discussed in more detail later in the paper). The first, and most important, is the identification of a thing with the content of its spatio-temporal extension. The second is the identification of a thing with the fusion of its time-slices. Both of these forms of identification require that the objects fuse in the manner of aggregation. It has also often been supposed that aggregation is the only legitimate method of fusion. Part of the appeal of this further position may arise from a general hostility to different methods of composition, whether they be methods of fusion or not. Under the form of nominalism championed by Goodman, for example, there can be no difference in objects without a difference in their parts; and this implies that the same parts cannot, through different methods of composition, yield different wholes. However, I suspect that many of those who would be open to structural methods of composition would still not be open to distinct nonstructural methods of composition. For it is hard to see, especially given the identification of a thing with its spatio-temporal content, what other methods of fusion there might be; and it is hard to see how there could be alternative conceptions of a fusion, of a whole at the same level as its elements and formed without regard to their order or repetition. Let us call the extreme position, that there is only one method of composition, mereological monism; let us call the less extreme position, that there is only one method of fusion, fusion monism; and let us call that particular version of fusion monism according to which aggregation is the sole method of fusion aggregation monism. The main purpose of this paper is to show that the last of these three positions is mistaken. I want to show that there is a method of fusion which is not aggregative, i.e. which does not conform to the characteristic existence conditions for aggregates. However, my attack on this position may be relevant to the two other positions as well. For granted that aggregation is itself a legitimate method of fusion, it follows that fusion monism should be dropped in favour of a pluralist position. And to the extent that the adoption of monism depended upon a general hostility to structural considerations, the way is then open to the admission of structural methods of composition. It is also my intention to attack two related forms of monistic doctrine. For just as we can single out the aggregative method of nonstructural composition, so we can single out the aggregative way of being a nonstructural part and the aggregative kind of nonstructural whole. One might then maintain that not only does aggregation constitute the only nonstructural method of composition, but that it also constitutes the only nonstructural way of being a part and the only nonstructural way of being a whole. We therefore have three forms of monism, one with respect to composition, another with respect to part, and a third with respect to whole. As will later become clear, the two further forms of monism are

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successively weaker than the original; and so their denials might be taken, in mimicry of Quine, to comprise three grades of mereological involvement. From the discussion of monism will emerge objections to two other prominent doctrines: extensionalism and mereological atomism. According to the first of these, things are the same when their extensions (spatial, spatio-temporal, or modal-spatio-temporal) are the same; and according to the second, parts are prior to their wholes. For the purposes of attacking the aggregation monist, I have assumed that aggregation is a legitimate method of fusion. Towards the end of the paper, I suggest that there is no such method and propose a form of fusion monism in which some other method of fusion takes the place of aggregation. However, my tentative endorsement of fusion monism is not meant in any way to lend support to a general monist position. $1 The First Grade of Mereological Involvement. Our central argument is based on a very simple example and I suspect that it is only through some sort of theoretical blindness that its signifance has not been recognized before. Let d and e be two quantities of matter, say of gold; and let c be the larger quantity of gold composed of d and e. Now I take it that c is formed from d and e in a certain way; it is the result of applying a certain operation, which we may call compounding, to the given objects. Thus using * to signify this operation, we have c = d * e. I wish to claim that compounding is a method for fusing parts into wholes which is not aggregative, i.e. which does not conform to the characteristic existence conditions for aggregates. Granted that aggregation is a legitimate method of fusion, it follows that compounding is a distinct method and hence that fusion monism is false. Our claim divides into two subclaims: first, compounding is a method of fusion; and second, it is not aggregative. For the purpose of establishing the second subclaim, we need merely suppose that the quantity d is destroyed, or otherwise ceases to exist, while the quantity e continues to exist (d is a mnemonic for "destroyed", e for "endures"). It is then apparent that the compound c, the larger quantity of gold, ceases to exist when the component part d ceases to exist; and consequently, compounding does not conform to the aggregative existence conditions, which require that the fusion should exist when only one of the components exists. It is also apparent, in the envisaged situation, that the aggregate a = d + e (where ' + ' here represents the operation of aggregation) will be distinct from the compound c = d * e. For a will exist at the time at which e exists without d; but c will not2. From this it follows that compounding is not just a trivial variant of aggregation in the sense that any application of compounding to given objects can be explained as the single or successive application of aggregation to those objects3. They are essentially different operations.

It is hard to see on what basis the existence of the compound c might be denied. For the compound is just another quantity of gold and there is therefore as much reason to suppose it exists as its smaller companions. Indeed, it seems to me that, for this reason, our intuitive judgement concerning the existence of the compound is more secure than our corresponding judgement concerning the existence of the aggregate. However, some philosophers have been troubled by the very idea of a quantity of matter. For they have supposed that any material thing must always be of a piece, whereas a quantity of matter is capable of being scattered. I myself would have no doubts on this score; and nor would any advocate of aggregates. However, there is no need to confront the issue. For the example can be conceived in terms of adjoining pieces rather then quantities. Thus d and e can be taken to be two adjoining pieces of gold which comprise the larger piece c. Given the existence of the compound c, it is also hard to see on what basis it might be denied that c ceases to exist when d ceases to exist. For surely a compound thing exists only when all of its components exist. This is especially clear in the case of compounds of quantities of matter. For such a compound is just another quantity of matter; and a quantity of matter exists only when all of the matter comprising the quantity exists. There may perhaps be a loose sense in which we can talk of a quantity of matter surviving even though not all of it survives. This may be a looseness either in the sense of quantity or in the sense of survival (or perhaps in both). I myself have no objection to such loose senses; I am happy to be as loose as the next philosopher. I merely insist upon the right to be strict; for it is only on the legitimacy of the strict senses, and not on the illegitimacy of the loose ones, that the argument depends. In any case, a version of the argument can be made to go through under a loose construal of what it is for a quantity of matter to survive. For no assumption was made about the relative size of the components d an e. So let us suppose that d is enormous and e is tiny. Then I take it that as long as d is enormous enough and e is tiny enough, it will follow, even under the loose construal, that the compound quantity c will not be capable of surviving the destruction of d. Nor is there any real alternative to the proposed existence conditions for aggregates. For the proponent of aggregates will want to say that an enduring thing is the aggregate of its time-slices. Now this special case conforms to the proposed conditions: the enduring thing exists just when one of its time-slices exists. But from the special case the general case follows. For any component of an aggregate is itself an aggegate of time slices; and so any aggregate of the components is also an aggregate of the time-slices of those components. Now the aggregate will exist just when one of its time-slices exist; and since each component will exist just when one of its time-slices exist, the aggregate will exist just when one of the components exists. Thus commitment to "slicing", the view that

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each enduring thing is the aggregate of its time-slices, requires commitment to the proposed conditions for existence at a time. We thus see that there is a marked contrast in the principles governing the temporal and spatio-temporal extension of compounds and of aggregates. In one respect, to be sure, these principles are similar. For should either a compound or an aggregate exist at a given time, its spatial extension at that time will be determined in the same way; it will be the union of the spatial extensions of its components at the time. But in another respect, the principles differ. For whether a compound or an aggregate exists at a given time will be determined conjunctively in the one case and disjunctively in the other: the aggregate will exist when one of its components exists; and the compound will exist when all of its components exist. We must now consider the question of whether compounding is a method of fusing parts into wholes. If compounding is a method of composition, then there is no doubt that it is nonstructural, i.e. a method of fusion. Moreover, if the supposed components of a compound are parts of the compound then there is no doubt that the compound is composed of them. So the only real question is whether the supposed components are parts of the compound. But how can this be denied'? Surely this larger quantity (or piece) of gold has the two smaller quantities (or pieces) of gold as parts. Do we not have here a judgement about part which is as firm as any? All the same, many philosophers would demur. Their misgivings can perhaps be traced to two main sources: one is the adoption of the 'content model'; and the other is the 'spatializing' of time or, what is similar, the 'eventifying' of things. Let us consider each in turn. According to the content model, a thing is to be identified with the appropriate content of its extension, with what it is in the thing that fills the extension. The extension may be regarded as some sort of container or bag and the content as what occupies the container. Or to be more graphic, we might represent the extension by a circle in a Venn diagram; the content would then correspond to what was in the circle. There are different versions of the model according to how broad one takes the extension of something to be; it could be spatial, spatio-temporal, or spatiotemporal-modal (where the modal extension is constituted by the worlds in which the object exists). And there are also different versions of the model according to how one conceives of content; it could be constituted by some sort of atoms or particles, for example, or by some sort of atomless stuff. The version of the content model of interest to us takes the relevant extension of a thing to be its spatio-temporal extension; and in order to fix the view, we may suppose that it takes the content which occupies such an extension to consist of the various mereological particles of the thing, where these are not its timeslices but the further restrictions of those slices to the points in space.

Under this mereological conception of the content model, it is hard to see how something of greater temporal extent could be part of something of smaller temporal extent; for the thing with greater temporal extent will have more content. The container will be bigger and there will therefore be more in it. It is interesting to note that an analogous objection arises for the orthodox advocate of aggregates, one who believes that an aggregate can only exist in a world in which its components exist. For why should we not identify a thing with the content of its spatio-temporal-modal rather than its spatio-temporal extension? We would then face an exactly parallel case; for there are worlds in which Socrates exists and Plato does not, and so Socrates would be part of the aggregate of Socrates and Plato even though his modal extension was greater. However, it is somewhat unclear, either in regard to compounds or aggregates, exactly what the difficulty is. Perhaps it is thought to arise from the fact that it is not evident from the mereological conception how the one thing can be part of the otheI.4. But even granted that our judgements of part-whole should be evident from our conception of the things in question, there is still no need to consider ourselves bound by one conception of the content model to the exclusion of another. Indeed, it seems clear that in the present case we could, with equal justification, adopt a materialistic conception of the content model. According to this conception, the relevant extension would be the spatial extension (at any given time at which the thing existed); and what occupied the spatial extension would be the thing's material particles, i.e. the particles in the ordinary scientific sense of the term, be they atoms or something more basic. But on such a conception it would be evident how the smaller quantity of gold could be part of the larger quantity, for it would contain fewer material particles. But perhaps the objection is that the two conceptions are incompatible. One difficulty in adopting both conceptions goes as follows. Take the enduring component e of the compound gold c. Take now a mereological particle p of the component e which exists at a time at which the gold c does not. Then the particle p, which seems completely foreign to the gold c, is part of the gold. But how can this be? Now in this case, in contrast to the previous one, it is not evident from the materialistic conception, or from any conception whatever, how the particle can be a part of the compound. But this is because both conceptions are involved. For the particle is only indirectly part of the compound, it is only part of the compound through being part of the component: and to see that the particle is a part of the component we must use the mereological conception; and to see that the component is a part of the compound, we must use the material conception. However, we should not be bothered by the fact that no single conception is adequate to the case. For given that certain conceptions justify us in making direct judgements of part-whole, the successive application of those conceptions will justify us in making indirect judgments. Given that the component e is a part

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of c according to the materialistic conception and that the particle p is a part of c according to the mereological conception, it will follow that p is a part of c according to the combined conception. And there is no reason in general to suppose that those judgements of part-whole which have a justification in terms of two or more conceptions should also have a justification in terms of one. Nor should we be bothered by the fact that the given part, the particle, is temporally disjoint from the compound. For the requirement of temporal overlap is something which has its basis in one or other of the two conceptions5; and we should not expect it to be preserved when the two conceptions are combined. (A somewhat similar case concerns the constituents of events. No one should deny that Caesar is a part of his death on the grounds that his initial temporal slice would then be a temporally disjoint part of his death). We see that the first kind of misgiving, based on the mereological conception of the content model, cannot be maintained. We turn now to the second kind of misgiving against the intuitive judgement that a component is part of a compound. This is based on the presumption that there should be a certain analogy between time and space. One cannot get more gold in less space (at a given time). So how, it may be asked, can it be possible for there to be more gold in less time. If the spatial extension of some gold can never, at a given time, be less than that of a part, then how can the temporal extension of some gold be less than that of a part? There is a related misgiving based upon the relationship between things and events. For just as it might be thought that the mereological behaviour of things should be the same with regard to space and time, it might be thought that the mereological behaviour of things and events should be the same with regard to time. Now an event cannot take more time than an event of which it is a part; some thunder, for example, cannot take more time than the thunder and lightning of which it is a part. So how can a thing "occupy" more time than a thing of which it is a part? It is somewhat unclear how to take these arguments; for the presumption that there is the required analogy between time and space or between things and events can have very little force when set against our strong intuition that this smaller quantity of gold is part of the larger quantity. Perhaps the most charitable way of taking the arguments is that the purported disanalogies stand in need of explanation and that the lack of any explanation is therefore a reason for believing them not to exist. I myself am disinclined to deny an apparently evident fact on the grounds that it cannot be explained; I have much greater confidence in my powers of intuition than in my ability to explain. But we may note that whatever force these considerations may have against the proponent of compounds, a related consideration will tell against the orthodox proponent of aggregates. Just as the one faces a disanalogy between time and space, the other faces a disanalogy between time and worlds. For even though he allows the aggregate of Socrates and Plato to

exist at a given time if either of them exist at that time, he requires both Socrates and Plato to exist at a given world if their aggregate is to exist at that world. Of course, he could remove this particular difficulty by adopting a "fivedimensional" conception of aggregates under which the aggregate of Socrates and Plato, let us say, would exist in a world in which only one of them existed. But the view would be very odd; it would have as a consequence, for example, that the aggregate of all the possibilia which existed in a possible world would exist in the actual world as long as just one of them did. And, in any case, the five-dimensionalist would still be left with a disanalogy between things and events. For a conjunctive event (such as thunder and lightning) exists in a world only if all of its component events (the thunder and the lightning) exist in the world. To avoid the disanalogies completely, he would also have to adopt a fivedimensional conception of fusion for events and would therefore run into conflict not only with the way things compound and aggregate but also with the way events fuse. Morever, even if one is prepared to deny the apparent facts about how objects fuse, it is still not clear how uniformity is to be achieved. One could indeed adopt what I called a five-dimensional conception of aggregation. But could one not, with equal justice, model the general behaviour of aggregates on the apparent behaviour of compound things with respect to times or of aggregates and compound events with respect to worlds? Instead of making the existence conditions uniformly disjunctive, one could take them to be uniformly conjunctive, with the extension of a fusion now being the intersection, rather than the union, of the extensions of its components. After all, one might think, how can a genuine composite be at a given place or time or world unless all of its components are there. Such a stand flies in the face of the apparent facts concerning the location of compounds and the like. But why should these facts be relinquished in favour of those concerning the duration of compounds or the modal dimensions of compounds and aggregates? It therefore appears that uniformity can only be achieved at the double cost of being extreme in regard to how much is denied and arbitrary in regard to what is denied. In addition, the assumption which renders uniformity desirable, viz. that the disanalogies lack explanation, is not altogether justified. For it is common to distinguish between things and events on the grounds that the former must be completely present at any time at which they exist whereas the latter need only be partially present at any time at which they occur. This distinction has usually been introduced to explain how things are capable of change (with respect to position or parts or properties) while events are not. But it may also be used to explain the present disanalogies. For if a compound thing is to exist at a time, it must be completely present at the time and this surely requires that its components be completely present, and hence exist, at the time. On the other hand, the occurrence of a compound event at a given time only requires its partial presence at that time and hence only requires the existence of one of its components. We

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can in a similar way explain the other disanalogies: the conjunctive behaviour of compound things, compound events and aggregates with respect to worlds on the ground that their complete presence is required in any world in which they exist; the disjunctive behaviour of aggregates and compound events with respect to times and of all of the fusions with respect to space on the grounds that only their partial presence is required at the relevant times or places; and the disanalogy between time and space on the ground that the complete presence of a thing is only required at the times, and not at the places, at which it exists or is located. It must be admitted that these explanations do not go far enough. For we lack a clear understanding both of what it is to be completely present6 and of why it is in general impossible for an object to be completely present at a partial location or partially present at a world. But all the same, the explanations do have considerable intuitive force and they do serve to relate the disanalogies to a broader range of phenomena; and to this extent, they make it seem much less mysterious why things should fuse in the manner required by compounding.

92 The Second Grade. We turn now to the second grade of mereological involvement. It will be argued, not merely that aggregation and compounding constitute different ways of composing wholes from parts but also that they correspond to different ways of being a part. Let us make the further claim more precise. A notion of part may be defined in terms of aggregation. For we may say that p is a part of w just in case w is the result of aggregating p with some object q . A similar definition of part can be given in terms of compounding; p is a part of w just in case w is the result of compounding p with some object q. To keep these two notions separate, let us call the first a-part and the second c-part7. Our claim is that the component e is not an a-part of the compound gold c. It is clear from how c has been defined that e is a c-part of c. Thus if the claim is correct, it follows that parts with respect to compounding are not always parts with respect to aggregation; to be an aggregative component is not the only way to be a part. This claim, that e is not an a-part of c, goes beyond our previous conclusion that the compound c = d * e is distinct from the aggregate a = d + e. For all that the latter requires is that c is not the result of aggregating e with d. But what we now need to know is that c is not the result of aggregating e with any other object. We may illustrate the difference between the two positions by means of a somewhat artificial example. Suppose that ordered pairs are a certain kind of structured whole. There are then two methods by which they might be composed: one composing the pair (x,y) from the objects x and y; and the other composing the pair (y,x) from the objects x and y. The corresponding senses of part are then the same (indeed, to a highly intensional degree) while the operations are dis-

tinct. Of course, in this case the difference between the two operations is trivial, it being merely a matter of the order in which the arguments are received (and nor is it clear that there is a better example, one in which the operations are distinct and yet are not plausibly taken to be variants of a single underlying method of composition). An extension of our original argument that aggregation is distinct from compounding may readily be given to establish the present claim. For let us suppose that e is an a-part of c. Then c is of the form e + f for some object f. So c will exist whenever e does and, in particular, will exist at the time at which e exists but d does not. But we know that c does not exist at that time. A contradiction. It is worth noting that a similar argument establishes that d is not a c-part of a. For otherwise a is of the form d * f for some object f. So a only exists when d does and, in particular, does not exist at the time at which e exists but d does not.

93 The Third Grade We come to the final grade of mereological involvement. It will be shown, not merely that aggregation and compounding correspond to different ways of being a part, but also that they correspond to different kinds of nonstructural whole. The claim that c and a are different kinds of whole goes beyond our previous one that e is not an a-part of c. For the latter claim only requires that c not be the result of aggregating e with something else. But what we now need to know is that c cannot properly be regarded as the result of aggregating any objects, whether these include e or not. In this case, we may find a more interesting illustration of the difference between the two grades. For let us suppose that set-theoretic union (taking us from sets x,, x,, ... to their union x, U x, U ...) and the set-builder (taking us from elements x,, x,, ... to their set {x,, x,, ...)) are both methods for forming sets from their parts. The senses of part which correspond to these operations are then different; for union will correspond to inclusion and the set-builder will correspond to the ancestral of membership (which holds between x and y when they are linked by a chain of membership relationships). However, the kinds of whole corresponding to these operations will be the same, viz. sets. Turning to the present case, how might it be maintained that the compound quantity of gold c = d * e is in reality an aggregate? What could it be an aggregate of? Certainly not of d and e or of e and anything else, as we have seen. There is, however, another possibility. Let d' and e' be the respective restrictions of d and e to the times at which both exist. Then it might be maintained that c is identical to the aggregate c' = d' + e'. There is no possibility of arguing against this proposal on the grounds of a difference in spatio-temporal extension, since d' and e' have been chosen so that the extensions of c and c' shall be the same. Whether there could be a possible discrepancy in the spatio-temporal extensions of c and c' is not so clear, since the modal behaviour of the restrictions d' and e' has not been specified. But we may

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suppose, in the spirit of the proposal, that possible spatio-temporal discrepancies can be avoided by defining d' and e' as the appropriate modal-temporal restrictions of d and e. There are those who would argue against the view that c = c' on the grounds that the required segments d' and e' do not exist. I myself would not wish to deny the existence of temporal or of modal-temporal segments. But we may observe, all the same, that it is odd that the two issues should be connected in the way that is being presupposed. For on the face of it, the existence of compounds does not depend upon, and indeed is much less problematic than, the existence of temporal or of modal-temporal segments. There is, however, another argument against the view, based on considerations not of extension but of part. If it is true in the given case that c = d' + e', then we may take it to be generally true that such an identity holds. For on what basis could it be true in some cases and not others? Now let But be a particular butterfly and Cat the caterpillar from which it descends; and take d to be But and e to be the aggregate Cat + But of Cat and But. The compound c = d * e is then, on the given proposal, identical to the aggregate c' = d' e'. Here d' and e' are the restrictions of d and e to the period of time during which they both exist, i.e. to the life-time of But; and so each is identical to But. So c' = But + But = But; and since the aggregate e = Cat + But is a part of c and c = c', it follows that the aggregate Cat + But is a part of Buts. I am not sure that there is anything formally wrong with such a notion of part, at least when considered on its own. Rather surprisingly, it turns out to satisfy the three fundamental requirements of reflexivity, transitivity and asymmetry. But such a notion is completely at odds with our intuitive understanding of what it is to be a part. How can the aggregate of Cat and But in any sense be a part of But? Indeed, in such a case we would like to say that But is a proper part of the aggregate. Not that this would in itself run counter to the conclusion that the aggregate is a part of But; for the aggregate is to be a part of But by way of compounding, while But is a part of the aggregate by way of aggregation. However, it seems reasonable to assume that no two ways of being a part will be incompatible with one another: if in one way the object d is a proper part of the object e then in no other way is e a part of d9. Under this assumption, it is then impossible for the aggregate of Cat and But to be in any way a part of But. We cannot therefore take the original compound quantity of gold c to be the aggregate d' + e'. But is there not another possibility? For let c' be the restriction of c to a proper subperiod of time during which it exists; and let c" be the restriction of c to the remaining subperiod of time. Then can we not maintain that c = c' + c"; and is not c therefore an aggregative whole? We here face the issue of what it is to be an aggregative whole or, more generally, of what it is to be a kind of whole which corresponds to a given method of composition. Now the obvious account is as follows: a is an aggrega-

+

tive whole if it is the result of applying the aggregation operator to certain objects; and similarly for other methods of composition. But such an account can hardly be what is intended. For under it, anythingeven a set-will qualify as an aggregative whole for the trivial reason that it is the result of aggregating itself (or of aggregating itself with itself, if it is thought that aggregation requires at least two arguments). Now there are two reasons why it might be thought that self-aggregation is not necessarily productive of an aggregative whole. One reason is that it is not necessarily productive of something which has a proper aggregative part (one distinct from the whole itself). To get a whole of a certain kind it must then be required that at least one of the composed objects be distinct from the result of the composition (which means, in the case of a nonstructural operation, that the composed objects must not all be the same). Accordingly, let us say that an object is a proper whole of a certain kind if it is the result of applying the given method of composition to some elements, at least one of which is distinct from the object itself. Under such a conception of proper whole, the compound c will be a proper aggregative whole. For the existence of one proper temporal (or spatio-temporal) restriction c' of c will guarantee the identity of c and c' + c. But there is another, more subtle, reason, why it might be thought that selfaggregation is not necessarily productive of an aggregative whole. For it might be supposed that in saying that something is an aggregative (or some other kind of whole) one is saying something about its analysis into parts. But no account of an object in terms of itself can provide an analysis of the object and can therefore, a fortiori, provide an analysis of the object into parts. In general, an analysis must be non-circular; the elements into which the object is analysed must not themselves be analysed in terms of that object. Now the circularity can be blatant, as when an object is analysed in terms of itself. But it can also assume a less blatant form; for even though the elements into which an object is analysed are distinct from that object, they may still require an analysis in terms of it. Thus it is conceivable that a proper whole of a given kind might be incapable of having any analysis as a whole of that kind. Accordingly, let us say in those cases in which a kind of whole is regarded as posterior to its parts, that an object is a genuine whole of that kind if it is the result of applying the given method of composition to some elements, no one of which need be analysed in terms of the object itself. Under such a conception, it is then no longer plausible to maintain that the compound c is an aggregative whole. For let us suppose, without any real loss in generality, that the proposed aggregative analysis of c is in terms of its time-slices. The question then is whether these time-slices must be analysed in terms of the compound c. Now there would appear to be only two ways in which one might understand what the time-slices are independently of the compound. One is to take a timeslice of c at a given time to be identical to the aggregate of the corresponding

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time-slices of d and e. But such an account would render c identical to the aggregate d' + e', which we know from our previous argument not to be the case. The other possibility is to treat the time-slices of the compound as just given. But this is metaphysically absurd. For we are being asked to believe that certain space-time locations are occupied by two momentary objects (the respective time-slices a,, a,, ... and c,, c,, ... of d + e and d * e)lO. But when we ask what makes these momentary objects distinct, no answer can be given. We are also being asked to believe that when we aggregate certain of these momentary objects (viz. c,, c,, ...) at different space-time locations, we obtain a compound, an object which is capable of having non-aggregative parts, parts of broader temporal extent than the whole. But when we ask what it is about the momentary objects, as opposed to others such as a,, a,, ... , which makes them capable of yielding wholes with such extraordinary properties, no answer can be given. Surely we have here a clear case in which the proper order of explanation has been reversed. We understand what the time-slices are in terms of the compound, not vice versa. The difference between the time-slices of d + e and d * e rests on the fact that one is a time-slice of d * e while the other is not; and what enables some of these time-slices to aggregate to a compound is the fact that they are the time-slices of the compound. We see that the two ways of analyzing c as an aggregate both fail. Since there is no reason to doubt that c has an analysis as a compound, we may conclude that aggregates do not, in the intended sense, constitute the only kind of nonstructural whole. But a worry may remain. For our whole discussion has been couched in terms of the notion of analysis; and many may wonder whether the notion makes any sense, especially in its application to objects. Now whatever difficulties might attach to the general notion of an analysis, I think that one can give a fairly clear account of what is involved in the present case. We may note, first of all, that we need only make use of the concept of one object presupposing another. Intuitively speaking, one object presupposes another if it both stands in need of analysis and is incapable of an analysis except in terms of the other. We can then define a (non-circular) analysis to be one which analyses any object in terms of objects which do not presuppose it; and we can define a genuine whole of a certain kind to be one which can be composed, in the prescribed manner, from objects which do not presuppose it. The concept of presupposition may, in its turn, be defined in terms of the concept of a given object. Intuitively speaking, the given objects are those which stand in no need of analysis (which is not to say that they are incapable of analysis). Thus for the mereological monist, the given objects may well be the "particles" which occupy the various points in space-time. However, there is no reason in principle why the givens should not include aggregates or compounds which are atomless in the sense that they contain no parts which are without

proper parts. So, for example, the givens might include time-slices which are arbitrarily small or temporal segments which are arbitrarily short or quantities of uniform matter which are arbitrarily divisible". From the given objects, new objects can be generated by successively forming aggregates or compounds or segments. An object d can then be said to presuppose another e if it is not possible to generate d without generating e'2. Thus according to these definitions, a genuine whole of a certain kind is one that can be introduced into the ontology as a whole of that kind; under some given manner of generation, the object makes its first appearance into the ontology as the result of applying the given method of composition to previously generated objects. Given this framework, our basic point concerning the inadequacy of aggregation can be rendered independent of any reference to the concept of genuine whole. It is that one cannot understand what nonstructural wholes there are in terms of aggregation (or of aggregation and segmentation) alone; under any reasonable construal of what is given, even allowing mereological particles or time-slices, there will be wholes which can only be introduced into the ontology as compounds. Thus compounding is required, not just as a way of discerning new parts within wholes that are already formed (as with the case of set-theoretic union), but also as a way of forming some of those wholes in the first place. 04 Against Atomism and Extensionalism. We have noted various differences between compounding and aggregation. I should now like to explore some consequences which flow from these differences. The first of these concerns the issue of holism versus atomism. Roughly speaking, mereological atomism is the view that wholes are to be analysed into parts. But there are various forms the doctrine can take, according as to exactly which analyses are required or forbidden. A relatively weak form requires that any object which has an analysis must have an analysis in terms of objects which are not wholes of which the given object is part. Such a view is, of course, compatible with an object having an analysis in terms of wholes which contain the given object as part as long as there is also an analysis which is not in terms of such wholes. Our account of compounds (along with the admission of time-slices) makes it plausible that this form of atomism is false. For as we have seen, the only plausible analysis of the time-slices of a compound is in terms of the compound itself. It should be noted that our argument against the analysis of a compound into time-slices is not based on any general presumption against the analysis of an enduring object into its time-slices. We may grant that ordinary thing which are not compounds are to be analyzed into their time-slices. The point rather is that the corresponding analysis of compounds gives rise to a special difficulty; for it

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requires an implausible extension of what would otherwise appear to be a plausible ontology. Nor does our argument rest on the assumption that compounds must be primitive, standing in no need of analysis. We may grant that compounds are to be analysed into their components; and we may even grant that all things are ultimately to be analyzed into mereological particles or the like. Our point is that this analysis will not always proceed from whole to part. Even if the compounds are analyzed in terms of their ultimate atomic components and even if these components are analyzed in terms of aggregative particles, the analysis of the time-slices of the compounds must be in terms of the compounds themselves. The second consequence concerns a doctrine which I hope may without confusion be called extensionalism. This doctrine roughly states that things are the same under the condition that their extensions are the same. But the exact meaning depends upon the sense of 'extension'. When the extension is spatial, the condition for identity is that there be a time and a world at which the spatial extension of the two things be the same. When the extension is spatio-temporal, the condition is that there be a world at which the spatio-temporal extension of the things be the same. And when the extension is spatio-temporal-modal, the condition is simply that this extension be the same. We thus arrive at three successively weaker forms of the view: the spatial, the temporal and the modal. One might think of extensionalism as arising from the content model and the further principle that the elements of the content, e.g. the particles, cannot coincide: for things with the same extension would then have the same content and so would be the same. However, the content model, when supplemented in this way, would appear to justify an extensional criterion of part (one thing being a part of the other when its extension is included in that of th other); and it is somewhat unclear whether the extensionalist with respect to identity would also want to be an extensionalist with respect to part. Many philosophers have had their doubts about the spatial or temporal forms of extensionalism. But even they have often been attracted by the modal form. They have supposed that there cannot be any difference between things without an actual or possible difference in their spatio-temporal extension. The space of possible worlds has been regarded as the ultimate repository of all differences. However, it seems to me that the acceptance of both aggregates and compounds renders the weak position untenable. We may show this by using our previous example concerning the caterpillar and butterfly. Recall that d is the butterfly and e is the aggregate of the butterfly and the caterpillar. The spatiotemporal-modal extension of d * e is then the same as that of d (whether one adopts a four-dimensional conception of aggregation or not). For if d * e exists in a world then so does d ; and if d exists in a world then so does the caterpillar and hence so does d * e; and in any world in which d * e exists, its spatio-temporal

extension will be the same as that of d . So modal extensionalism requires us to say that d = d * e, i.e. that e is a part of d; and, as we have seen, this is false. A similar difficulty may arise for someone who believes only in compounds. For the only conditions on e required for the argument to work are that (i) e not be a c-part of d and (ii) e exists at any time (and in any world) at which d exists and at any such time has the same spatial extension as d; and it is not clear on what grounds a modal extensionalist who endorses compounds could exclude the possibility of an object e satisfying these conditions. A similar difficulty may even arise for someone who believes only in fourdimensional (i.e. four-dimensional) aggregates. For an analogous argument will work with aggregation in place of compounding as long as e is subject to the conditions that (i) e is not an a-part of d and (ii) e exists at any world at which d exists and at any such world has the same spatio-temporal extension as d; and again, it is not clear on what grounds the possibility of such an object is to be excluded. In general, some rationale should be given for supposing that compounds or aggregates conform to the prescribed form of extensionalism. For appropriately strong forms, this may not be a problem. If, for example, the spatial form is in question, then one could suppose, on the basis of the appropriate content model, that the relationships of spatial inclusion corresponded to relationships of part. But the weaker forms (temporal or modal for compounds, modal for aggregates) are problematic. For the conjunctive element in the existence conditions for the respective whole is at odds with the extensional criterion of part, the one allowing the extension of a part not to be included in that of the whole and the other requiring that it be so included. Our example concerning the butterfly may lead one to wonder in what the difference between f = d * e and d might consist, given that their extension is the same. But the answer is plain; for f has e as a part while d does not. This difference in part is not, by hypothesis, manifest as a difference in extension. Nor does there seem to be any alternative, non-mereological, account of what the difference might consist in. We are thus forced to recognize differences between things of an ultimately mereological character. In this example, the difference between the objects d and d * e could be explained in terms of an undifferentiated notion of part. But under a modification of the example, the difference can only be explained in terms of the way the parts are parts. For, with d' = Cat and e = d + d' as before, consider the objects d * (d' + (d * e )) and d * e (obtained by letting d * e play the role of d in the earlier definition d * (d + d') of c). Then it is plausible that the two objects are distinct; for if they were the same, d' + (d * e) would be a proper part of d * e, which would, in its turn, be a part of d' + (d * e). The difference in the objects is not extensional; and nor can the difference be explained in terms of an undifferentiated notion of part since, if we were blind to the difference between + and *, we would also be blind to the difference between d * (d' + (d * e)) = d * (d' + (d *

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(d' + d)) and d + (d' + (d + (d' + d)) = e, on the one hand, and between d * e = d * (d' + d) and d + (d' + d) = e, on the other, and hence blind to the difference between d * (d' + (d * e)) and d * e. It therefore appears that the explanation of the difference must rest on the way the respective objects are composed of their parts. Of course, in the majority of cases, there will be an explanation of the difference between one aggregate-compound complex and another in terms of their extensions. But even in these cases one should still, in the interests of uniformity, view the explanation of the difference in terms of composition as more fundamental and the explanation in terms of extension as merely consequential upon it. What has perhaps stood in they way of the recognition of this natural point of view is the conception of things as the formless content of certain spatio-temporal or spatio-temporal-modal extensions. This conception may itself rest on a double standard in regard to abstract and concrete objects. For no one who accepted the objects in question would have any difficulty in grounding the difference between the set {a, b) and the ordered pair (a, b) in a difference in the way they were formed from their elements. But if structure can be essential in this way to the discrimination of objects in the abstract sphere, then why should it not be equally essential in the physical sphere?

$5 The divergence between compounds and aggregates. We have seen that some genuine compounds are not genuine aggregates. But when, if ever, are they the same? When is a compound of the form a * b, with a distinct from b, the same as an aggregate with an analysis of the form e + f? If it is supposed that a given compound is the same as an aggregate, there must be some general features of them in virtue of which they are judged the same. Now there would appear to be only two relevant types of feature: one having to do with the extension of the objects in question; and the other having to do with their structure, the way they are constructed. Now if one is guided exclusively by features of the first kind, then one must declare two things to be the same when their extension is the same. But, as we have just seen, once compounding and aggregation are both admitted as methods of composition, such a position cannot be maintained. If one is guided exclusively by reasons of the second kind, then presumably the particular judgment of identity must be based on some general algebraic law relating to the operations which are involved in the generation of the objects. But the obvious candidates for such laws lead to difficulties. Consider, for example, the following distributivity principle: (e

+ f) * g

=

(e

* g) + (f * g).

If such a principle held, it would seem to allow a genuine compound of the form (e + f ) * g also to be a genuine aggregate of the form (e * g) + (f * g); the object

could be introduced into the ontology either as a compound via (e + f) and g or as an aggregate via e * g and f * g . However, acceptance of the principle would lead to erroneous judgements concerning part. For (e f ) = (e + f ) * (e + f ) = (e * e) + (e * f ) + (f * e) + (f * f ) (by two applications of the distributivity law) = (e + f ) + (e * f); and so e * f is a part of e f. On the other hand, by another application of the distributivity law,(e + f ) * e * f = ( e * e * f ) ( f * e * f ) = e * f ; a n d s o e + f i s a p a r t o f e * f. But it then follows that e f = e * f, which we know is not generally so. In the above argument, the operations * and + play symmetric roles; and so by reversing these roles, we may show that the dual distributivity principle:

+

+

+

+

also cannot be accepted. It is possible that the basis for the identity judgement is partly extensional and partly mereological in character. For example, one might maintain that e * f is identical to e + f i n the special case in which the modal-temporal extensions of e and f are the same. If such a principle were adopted, one would have to allow that a compound e * f (with e and f distinct) might be identical to the corresponding aggregate e f, that a proper c-part (either e or else f ) might be an a-part, and that an object might be both a genuine compound and a genuine aggregate. Thus none of the three grades of mereological pluralism would hold in a thoroughgoing form. I myself am disinclined to accept the proposed principle; it smacks too much of extensionalism. But even with the principle, and others like it, the possibilities for identification still seem to be severely limited. In particular, it is hard to see on what basis one could identify the objects at one level in the successive application of aggregation and compounding with the objects at a lower level. For consider the following infinite sequence of objects: do = d, d, = d * (d' + do), dZ= d * (d' dl), d3 = d * (dl + d2), ... , where d is the butterfly and d' is the caterpillar. Then an extension of our previous reasoning shows that they are all distinct and hence belong to infinitely many levels13. Thus even though neither operation, when taken on its own, is capable of generating any new objects beyond the first level, the two operations, when taken together, are capable of generating new objects at each successive level.

+

+

96 The Legitimacy of Aggregation. It has been argued that there is a non-aggregative method of fusing things. But the question of whether there is also an aggregative method has not been considered. This must now be done. We cannot hope to survey all of the arguments which might reasonably be raised for or against the existence of aggregates. But we may note, once armed with the distinction between aggregates and compounds, that many of the argu-

COMPOUNDS AND AGGREGATES 155 ments which apparently favour aggregates may be better regarded as favouring compounds. Consider, for example, the argument to the effect that we ordinarily refer to aggregates, as in the statement 'the apples in this basket weigh two pounds'. One might well take the phrase 'the apples in this basket' to refer to a fusion. But surely it is much more plausible to suppose that the fusion in question is a compound, existing only when each of the apples exists, and not an aggregate, existing when only one of the apples exists. There appears, however, to be a very powerful argument in favour of aggregates. For there would appear to be no doubt that we fuse events by means of an operation which conforms to the disjunctive conditions for existence at a time. The fusion of thunder and lightning, for example, is an event which begins with the lightning and ends with the thunder. But if events can be fused in this way, then why not things? For surely the operation which is used to fuse events can also be used to fuse things; and if the operation is subject to disjunctive existence conditions in the one case, then surely it is subject to disjunctive existence conditions in the other case as well. It is difficult to see on what basis this argument might be criticized. It can hardly be denied that events fuse in the prescribed manner; and although events and things differ in kind, it is hard to see why this provides a ground for limiting the method used to fuse events to events alone; and, finally, it is hard to see how a single method of composition could be subject to differential existence conditions, working one way for one kind of object and another way for another kind. There is, however, something deeply suspicious about the argument, since an exactly analogous argument leads to counter-intuitive results. For it is apparent, or so the analogous argument goes, that we may fuse things by means of an operation which conforms to the conjunctive conditions for existence at a time, i.e. by means of compounding. But surely that operation which is used to fuse things can be used to fuse events; and surely it is then subject to the same existence conditions. But this is absurd. There is no way of fusing a sequence of lightning and thunder so as to get an event which exists when only both the thunder and lightning exist. There might, indeed, be an event which consists in the simultaneous occurrence of thunder and lightning. But this would, at best, have only part of the original thunder and lightning as its parts and would not, in the requisite sense, be their fusion. What has gone wrong? I would like to suggest that the error lies in the presupposition that events are temporal objects, i.e. objects which exist or are capable of existing at a time. With the presupposition, the view that events fuse disjunctively would be justified, as would the view that compounding extends to events. But without the presupposition, these assumptions are quite dubious; for if events are atemporal, then no condition for existence at a time is implicit in the manner in which they fuse and it is unclear how compounding, when it is defined in terms of its application to temporal objects, is to be extended to atemporal objects such as events.

What makes it so tempting to suppose that events are temporal objects is their sharing a certain feature in common with those objects which are temporal. For both events and things have, in an obvious sense, a temporal extension; and it is then easy to suppose that each, alike, exists at the different times within that extension. But for the purpose of understanding the atemporal status of events, it is probably better to make a comparison with periods of time, with "events" that are shorn, as it were, of any real content. For we are under no real temptation to suppose that periods of time are temporal objects; and it would appear to be in much the same way that events concern time without existing at a time. What is required to see matters aright is a general distinction between the dimension throughout which an object exists and the extension at which it is located. The extension, in this sense, will always be distinct from the dimension and, indeed, in a certain sense, orthogonal to it. Thus the dimension of things will be in time and the modal sphere, the extension in space; while the dimension of events will be in the modal sphere, the extension in time or space-time. What causes the confusion is the identification of the dimension of an object with its extension proper. Once this distinction is respected, the way is open to seeing the natural fusion of events (via the apparently disjunctive operation) and the natural fusion of things (via the genuinely conjunctive operation) as instances of the same method of composition. For this method, we may suppose, is subject to conjunctive existence conditions and distributive extension conditions. The "points" at which a fusion of this sort exist are those at which all of the components exist; and the points at which the fusion is located are those at which at least one of the components is located. Since things exist at different times, a fused thing will only exist when all its components do; and since events are located at different times, a fused event will be located at any of the times at which its components are located14. Acceptance of this point is compatible with believing in more than one method of fusion. For one might hold that, even though there was a single unitary method of fusing things and events, there was in addition an aggregative method of fusing things. Such a method would be necessary, for example, if one wished to construct enduring things from ephemera. However, acceptance of a unitary method of compounding makes a monist position much more plausible; for it is then able to accommodate the natural method of fusing things and of fusing events. Indeed, it might even be able to make room for something akin to aggregation. For one might suppose that to each object which exists in time there corresponds an object, its space-time worm, which is located in time. Thus rather than thinking of there being two alternative conceptions of the same category of objects, one three-dimensional and the other four-dimensional, we think in terms of these being two distinct categories of objects, each answering to its own conception. The aggregate of objects which exist in time could then be identifed with the compound of their

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corresponding worms. Another possibility (which I prefer but will not develop) is to define aggregation in terms of the more fundamental operation of embodiment. This operation takes any property (which is had by things at a time) into an object, called the embodiment of the property, which at each time is constituted by the things which have the property. The aggregate of the embodiments of P and Q is then identified with the embodiment of the property (P or Q). This monism is not of the usual sort; for it is monism with respect to compounds, and not with respect to aggregates. It strikes me, however, as being much more plausible than monism of the usual sort. For compounds of familiar things are familiar things-a cup and saucer, some apples, a quantity of gold. On the other hand, aggregates of familiar things, if they exist, are not familar things; and in that respect, their existence is open to doubt15

Notes here conceive of a method of composition cr as a polyadic operation which takes the parts into the given whole. Thus the condition concerning repetition says that u( ... , x, x, . . . , x, .. .) = u(... , x, ...) ;and similarly for the other conditions. If u is not allowed to apply to only one object, then the condition on a single object should be replaced with u(x, x) = x. ,The above argument for the distinctness of a and c rests upon the assumption that there is actually a time at which one of the components exists and the other does not. But one may also establish distinctness under the assumption that the envisaged circumstance is possible, i.e. under the assumption that in some possible world d and e exist and yet exist at different times. For in that world, the aggregate a' and the compound c' of d and e both exist; and they exist at different times by the same considerations as before. So, given that a = a' and c = c', a and c are distinct. This leaves the case in which d and e essentially have the same temporal extension, i.e. exist at the same times in each of the worlds in which they both exist. There is then no difference in the spatio-temporal-modal extension of the compound and the aggregate; and so not even any possibility of applying the modal argument. All the same, I am inclined to think for reasons which will later become apparent that in most, if not all of these cases, the aggregate and the compound will be distinct. 3Say that a polyadic operation u generates y from the objects x, , x,, ... if either (i) y is one of x , , x,, ... or (ii) y is of the form u(y,, y,, ...), where u generates y,, y,, .. . from x,, x,, ... . Say that u ' is a variant of u if, for any x,, x,, ... for which y = u'(x,, x,, . ..), cr generates y from x , , x,, ... . The claim, then, is that * is not a variant of + . It also follows, by symmetry, that is not a variant of *. Of course, in the case of a nonstmctural operation a , the objects generated from objects x,, x,, ... are of the form a(x,,, xi,, ... ) for some selection xi,, xi,, . . . of the objects x,, x,, and so any variant of u must be a restriction of a . But in general (for example, in the case of the set-builder), there will be many other variants. 4We might think of a conception as providing us with a set-theoretic representation of the things. The conception then makes evident those cases of part-whole in which the set represented by the part is included in the set represented by the whole. 51t is arguable that a component might not even temporally overlap with a compound. For consider some gold now and the compound of all gold, past, present and future. The gold now is then part of the totality of gold, but it does not overlap with the totality since there is no time at which the totality exists. This type of example is significant in other ways as well. For if it is accepted it shows that: (i) a paradigm physical object, viz. a quantity of matter, may exist and yet not exist at any time; (ii) two quantities of matter, e.g. all gold and all silver, may have the same spatio-temporal extension, viz. the null extension, and yet be distinct; and (iii) some quantities of matter are not aggregates, since otherwise the totalities of gold and of silver would both have to be identical to the null aggregate. Further interesting consequences arise from positing the null compound, an object whose compound with any object is that object itself and which is therefore required to exist at all times with

+

empty spatial location. I am inclined to accept these extreme cases; but I have not let my arguments rest on them. 61t is common to explain the complete presence of an enduring thing in terms of its not having any proper temporal parts. But surely some enduring things have other enduring things as temporal parts-a caterpillar, for example, is a temporal part of the insect. And if this is so, the proposed explanation of complete presence must be given up. 'Recall the definition of generation from endnote 3. We may then say that, for an arbitrary method of composition u , x is a u-part of y iff u generates y from some objects which include x. This definition then reduces to the one in the text given the fact that aggregation and compounding are nonstructural. &Itcan be shown, more generally, that x is a part of y if (i) x exists whenever y does and (ii) the temporal restriction x' of x to the times at which y exists is an a-part of y. For x is a part of x * y; and

*y

x'

+ y'

by the proposal, by (i), = y by (ii). 'It also seems reasonable to assume the compatibility of all the different ways of being a part: the ancestral of the union of these different ways should be antisymmetric. loin general, there will be many more momentary objects at a given space-time location. Suppose, for example, that there are three atoms of gold d , e, and f. Then we should probably distinguish betweenthetime-slicesofd*e*f,(d*e)+f,d+(e*f),(d*f)+e,d+e+f,(d+e)*f,etc., as well as the various spatio-temporal segments of larger compounds of atoms. "The quantity of matter is divisible in the sense of being divisible into c-parts, not a-parts. ',Let a derivation of an object y be a sequence x,, x,, ... , whose last element is y and in which each element is either a given or is of the form a(x,,, xi,, ...), where xi,, xi,, ... are previous members of the sequence and u is one of the given operations (in the present case, aggregation, compounding or segmentation). The object y is then said to presuppose x iff any derivation of y contains x. It should be noted that a given need not appear in a derivation as a given and hence may be a genuine compound or aggregate. I3In general, d,,, = d * (d' + d,). It is clear that d, is an (undifferentiated) part of d,,,. Suppose the two were the same. Then d' + d, would be a part of d,; and so, given that d, is a part of d ' + d,,, , d, and d' + d, would be the same. But it is readily shown that d, and d have the same extension. Hence d, and d' + d, have a different extension; which is a contradiction. Given that each d, is a proper part of d,, , it follows that the members of the sequence do, d l , ... are all distinct. To show that they belong to infinitely many levels, it suffices to assume that any generation from them must be from d and d' as the only givens and with aggregation and compounding at the only methods of composition. For at each finite level, there will be only finitely many objects which can be generated in this way. It is also plausible that each d, is introduced at level n and, for n > 0 , is a genuine compound. However, it remains to develop a systematic theory of the interaction between the operations of aggregation, segmentation and compounding, upon the basis of which such questions could be definitively resolved. I4Things and events have, at best, a location in space. This makes it natural to wonder whether there are any objects which exist through space. Odours are, perhaps, an example. Thus a compound odour will, in the required unitary sense, exist at exactly those places at which the component odours exists; and this is as it should be. Another indication that we are on the right track is given by the fact that there is no natural way of fusing an event with a thing. For when is the resulting entity to exist or be located? If events existed in time, there would be no more difficulty in fusing an event with a thing than in fusing two things. But if things exist in time and events do not, then the existence and extension conditions for unitary compounding will have no obvious application. I5I should like to thank Ruth Chang for her helpful comments on an earlier version of this paper. x

=

= x'

+y

,