Acta Analytica

, Volume 26, Issue 2, pp 115–133

Is There A Quasi-Mereological Account of Property Incompatibility?

Authors

Article

DOI: 10.1007/s12136-010-0090-0

Cite this article as:
Kalhat, J. Acta Anal (2011) 26: 115. doi:10.1007/s12136-010-0090-0
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Abstract

Armstrong’s combinatorial theory of possibility faces the obvious difficulty that not all universals are compatible. In this paper I develop three objections against Armstrong’s attempt to account for property incompatibilities. First, Armstrong’s account cannot handle incompatibilities holding among properties that are either simple, or that are complex but stand to one another in the relation of overlap rather than in the part/ whole relation. Secondly, at the heart of Armstrong’s account lies a notion of structural universals which, building on an objection by David Lewis, is shown to be incoherent. I consider and reject two alternative ways of construing the composition of structural universals in an attempt to meet Lewis’ objection. An important consequence of this is that all putative structural properties are in fact simple. Finally, I argue that the quasi-mereological account presupposes modality in a way that undermines the reductionist aim of the combinatorialist theory of which it is a central part. I conclude that Armstrong’ quasi-mereological account of property incompatibility fails. Without that account, however, Armstrong’s combinatorial theory either fails to get off the ground, or else must give up its goal of reducing the notion of possibility to something non-modal.

Keywords

David ArmstrongDavid LewisProperty incompatibilityReduction of modalityMereologyStructural universals

David Armstrong has propounded a combinatorial theory of possibility. This theory ‘traces the very idea of possibility to the idea of the combinations—all the combinations—of given, actual elements’, namely universals (properties and relations) and particulars (1989: 37; see also, 1986a, 1997).1 The theory is intended to be reductive of the notion of possibility, since the latter ‘is given an analysis, an analysis which uses the universal quantifier’ (1989: 47), i.e., which merely uses the universal quantifier. Combinatorialism faces the obvious difficulty, however, that not all properties are compatible.2 For example, if a particular a is red all over, then it cannot at the same time be green all over; if a particular b is exactly three meters long, then it cannot at the same time be exactly four meters long, etc. Armstrong is aware of this difficulty, of course, and by way of a solution proposes what I shall call the ‘quasi-mereological’ account of property incompatibility.

In this paper, I examine the quasi-mereological account, and develop three objections against it. First, Armstrong’s account cannot handle incompatibilities holding among properties that are either simple, or that are complex but stand to one another in the relation of overlap rather than in the part/ whole relation (Sect. 2). Secondly, at the heart of Armstrong’s account lies a notion of structural universals which, building on an objection by David Lewis, is shown to be incoherent (Sect. 3). I consider and reject two alternative ways of construing the composition of structural universals in an attempt to meet Lewis’ objection. An important consequence of this is that all putative structural properties are in fact simple. Finally, I argue that the quasi-mereological account presupposes modality in a way that undermines the reductionist aim of the combinatorialist theory of which it is a central part (Sect. 4). I conclude that Armstrong’ quasi-mereological account of property incompatibility fails. Without that account, however, Armstrong’s combinatorial theory either fails to get off the ground, or else must give up its goal of reducing the notion of possibility to something non-modal.

1 The Quasi-Mereological Account of Property Incompatibility

Properties typically come in ranges, forming classes of determinates of a determinable. The properties of being exactly one meter long, being exactly two meters long and being exactly three meters long, for example, are three determinates of the determinable length. Armstrong claims that ‘every such class of determinates falling under a determinable is held together by partial identities’ (1997: 52). Partial identity is the relation of overlap, which includes the relation of part to whole. Thus, the conjunctive property P&Q and the property P are partially identical, and so are the conjunctive properties P&Q and Q&R.3

According to Armstrong, then, the properties of being exactly one meter long, being exactly two meters long, and being exactly three meters long are partially identical. More specifically, being exactly one meter long is a proper part of being exactly two meters long, and being exactly two meters long is in turn a proper part of being exactly three meters long. And what goes for length also goes for other ‘remarkably unified classes of determinates’, such as mass, volume, duration, etc (1997: 55).

Such determinates are also structural properties. A structural property is a property that has parts, and which is instantiated by complex particulars the parts of which instantiate the parts of the property (1978: Chap. 14–5; 1989: 79; 1997: 26–8, 31–8). Thus, if a particular a instantiates the property of being exactly two meters long, then it follows that a proper part b of a instantiates the property of being exactly one meter long, itself a part of the property of being exactly two meters long.

Now consider the incompatibility that holds between, say, the properties of being exactly five kilograms in mass and being exactly two kilograms in mass. These properties are partially identical; in particular, being exactly two kilograms in mass is a proper part of being exactly five kilograms in mass. And they are also structural properties. Thus, any particular a which instantiates being exactly five kilograms in mass will have a proper part b which instantiates being exactly two kilograms in mass. But if so, to attribute both properties to a amounts to identifying a with b, that is, it amounts to identifying the whole with one of its proper parts (1989: 79; 1997: 54, 144–5). For in order for a to be two kilograms in mass, a would have to be that part of itself which is two kilograms in mass, namely, b. Therefore, if an object has the larger value of mass, then the only particulars that can have the smaller value are its proper parts (1989: 79). In this way, then, Armstrong explains the phenomenon of property incompatibility in terms of the principle that a whole cannot be identical with one of its parts.4 This, in essence, is Armstrong’s account of property-incompatibility. I shall call it Armstrong’s ‘quasi-mereological’ account of property-incompatibility—‘mereological’ because it trades on the part/ whole relation, and ‘quasi’ to remind ourselves that, as already noted, Armstrong takes complex properties to be something more than just mereological sums. I will now develop three objections against Armstrong’s account.

2 The Challenge Posed by Simple and Overlapping Properties

A first difficulty for Armstrong’s account is that there are cases of property incompatibility which it cannot accommodate. As we have seen, the account requires that incompatible properties stand in the part/ whole relation to one another. But while this may be true for properties such as being three meters long and being four meters long, it is not true for all such properties. The account will not work, in particular, for two kinds of resembling but incompatible properties: simple properties, and properties that are complex but stand to one another in the relation of overlap rather than in the part/ whole relation. The first kind of property includes positive electric charge and negative electric charge, which Armstrong himself offers as plausible candidates of simple properties (1989: 40). Lewis suggests that Armstrong might treat the incompatibility here as merely nomological, grounded in a contingent law of nature rather than in the nature of the properties themselves (1992: 197–8). Perhaps it could be discovered that neutral particles really have both charges, which somehow cancel each other out. Positive and negative charge are determinates of the determinable charge, however, and incompatibilities between determinates of a determinable are supposed to hold of absolute necessity. By letting negative and positive charge be metaphysically if not nomologically compatible, therefore, Lewis’ suggestion opens the door to a parallel treatment of incompatibilities holding among determinates of other determinables. Might it then be metaphysically possible for an object to be, say, uniformly round and uniformly square at the same time, etc? Surely not!

Armstrong tackles the difficulty posed by positive and negative charge in his discussion of ‘intensive properties’. Intensive properties are properties whose values do not depend on the ‘size’ of the particulars that have them. Two samples of a liquid, for example, can have the same temperature or density, even though one has a volume of one cup and the other fills a swimming pool. Thus, it is not the case that if a certain quantity of (say) mercury is 13.6 g/cm3, then a proper part of that quantity is 10 g/cm3, and another part is 7.4 g/cm3, and so on. Intensive properties pose a natural difficulty for the quasi-mereological account. A quantity of mercury that is 13.6 g/cm3 does not have a proper part that is (say) 10 g/cm3, and yet the original quantity cannot be both 13.6 g/cm3 and 10 g/cm3. The quasi-mereological account appears to have no purchase on incompatibilities among determinates of intensive determinable properties.

By way of solution, Armstrong suggests that density ‘can be resolved into, and so is supervenient on, volume and mass’ (1989: 80). Since both volume and mass are extensive quantities—i.e., quantities whose values do depend on the ‘size’ of the particulars that have them—the quasi-mereological account can perhaps deal with density-incompatibilities after all. Armstrong suggests that charge might similarly be reduced to extensive properties of particles. He writes:

Charge gives rise to attractions to and repulsions from other charged particles. These attractions and repulsions in turn give rise to motions or lack of motion. Motion and rest, however, can be cashed in terms of distances and times, which are extensive quantities. The charge itself could be treated as a mere disposition of the charged particle to attract and repel other particles according to certain formulae (1989: 81).

Since Armstrong rejects ‘bare dispositions’, he thinks that charge must be grounded in a categorical property, which together with the relevant laws of nature determine the behavior of the charged particles. To avoid regress, however, the categorical property must be an extensive property. As Armstrong is fully aware, this means that charged particles must be complex particulars. This is a significant encroachment on science, which as a naturalist Armstrong should find particularly unwelcome.5 For science takes charge to be a property of fundamental particles, and I take it that their fundamentality is at least partly a function of their simplicity. They are fundamental in that everything is composed out of them, while they are not themselves composed out of anything else. Of course, science might discover that fundamental particles such as electrons and protons are not in fact truly fundamental. Indeed, according to one currently popular theory, the truly fundamental building-blocks of the universe are vibrating strings (see Greene 2000). But this lends no support to Armstrong, for the point remains that it takes science to establish that so-called fundamental particles are not simple, and hence not fundamental. Armstrong is not entitled to that finding on the basis of purely a priori considerations. Furthermore, strings will themselves have properties, and what guarantee is there that none of those properties will be determinates of an intensive determinable property? If some are, then Armstrong will not be able to account for their incompatibility via a reduction to extensive properties—not unless he is then prepared to contradict the scientist over the fundamentality of strings.

The second kind of properties that poses a difficulty for Armstrong’s quasi-mereological account are those that are complex but stand to one another in the overlap rather than in the part/ whole relation. Shape properties, e.g., being a square, being a triangle, etc., are a case in point. Such properties give rise to an important class of property incompatibilities, since nothing can be both a square and a circle, a triangle and a rectangle, and so on, for any two different shapes. Thus, an important test for the quasi-mereological account is whether it can accommodate the phenomenon of ‘shape incompatibility’. Armstrong does not address the issue directly. A straightforward application of the quasi-mereological account, however, would presumably go something like this.

What we want to explain, suppose, is why a figure a, which is a triangle, cannot also be a square. So the explanandum is
$$ a\;{\hbox{is triangular}}\; \to \;\neg \;a\;{\hbox{is square}} $$
(1)
(where ‘→’ is understood as entailment). Armstrong could say that the explanation for this incompatibility is that there is a figure y which is a proper part of x and which is square, and accordingly that the attempt to combine the two properties would amount to identifying the whole with one of its proper parts. So the explanation runs
$$ \exists y\left( {y\;{\hbox{is a proper part of}}\;a\& y\;{\hbox{is square}}} \right) $$
(2)
(2) is perfectly correct. Any extended figure of any shape will have proper parts of any other shape. But notice that it cannot provide an explanation of (1). For
$$ \exists y\left( {y\;{\hbox{is a proper part of}}\;a\& y\;{\hbox{is triangular}}} \right) $$
(3)
holds for the same reason, yet it is obviously not the case that
$$ a\;{\hbox{is triangular}}\; \to \;\neg \;a\;{\hbox{is triangular}} $$
(4)
Therefore (2) cannot be what explains (1). The straightforward application of the quasi-mereological account to shape incompatibility fails.

As I said a moment ago, Armstrong does not directly address the issue of shape incompatibility. However, he does consider the question of whether shapes are partially identical.6 And this is, indeed, a vital step. For only if shapes stand in the part/ whole relation can the quasi-mereological account have a purchase on their incompatibility. The trouble is that unlike the properties discussed so far (viz., mass, length, etc.), the quantitative nature of which might be taken to readily establish a relation of partial identity holding between them, it is unclear that shapes are partially identical.7 Before I consider Armstrong’s account of the partial identity of shapes, however, I need to briefly introduce an element of Armstrong’s conception of properties not mentioned thus far.

Armstrong construes properties as state-of-affairs types (1997: 28–29). A state-of-affairs type is what remains of a state of affairs when all its constituent particulars have been abstracted away in thought. Thus, if a’s being F is a state of affairs, then something’s being F is a state-of-affairs type. Furthermore, Armstrong construes complex properties as conjunctions of states of affairs types (1997: 34–37).8 For example, the property of being exactly two meters long is identical with the conjunction of two state-of-affairs types, each of which is something’s being exactly one meter long.

We can now return to Armstrong’s account of the partial identity of shapes. This is what he proposes:

What is it to be a triangle? It is to be a thing enclosed by boundaries having just three parts, each of which is a straight line. Here we have three non-overlapping particulars, each of which has the properties of being straight and being a line [...] lines involve length and that is a determinable of a type that I have claimed to give an account of. These three lines are related to each other. Each of the lines meet the other two at their end points and there form an angle. There seems nothing here that cannot be spelt out in terms of properties of the three boundaries and relations of the three boundaries to each other. This can be spelt out as a conjunction of state-of-affairs types involving the three parts. What is it to be a quadrilateral? Here four boundaries are required, again they are straight lines, and they are connected by the lines meeting at an angle at end points, the lines being linked up in cyclic and non-intersecting fashion. The difference between the two shapes lies in the difference between three straight lines and four straight lines (which ensures that a triangle cannot be a square) while the resemblance lies in the fact that the parts involved are three and four straight lines, lines which are connected up according to the same formula (1997: 56).

Armstrong’s suggestion, then, is that shapes are complex properties, and hence conjunctions of state-of-affairs types. The property of being a triangle, for example, involves the conjunction of three state-of-affairs types, each of which is of the kind: some particular or other having the properties of being a line, being straight, and having a length. The property of being a quadrilateral, on the other hand, involves the conjunction of four state-of-affairs types of that same kind. Thus, the property of being a triangle and the property of being a quadrilateral are partially identical.

But are the two properties partially identical in the sense of part/whole or in the sense of overlap? It had better be in the sense of part/whole, for only if it is shown that a triangle is a proper part of a quadrilateral, or the quadrilateral a proper part of the triangle, will the quasi-mereological account have a purchase on their incompatibility. At the same time, it is difficult to see exactly how shapes stand in the part/ whole relation to one another. I think there may in fact be two different proposals in Armstrong’s account as to how shapes stand in the part/ whole relation to one another. On the one hand, a proposal in terms of the proper parts of shapes, and on the other, a proposal in terms of the properties of those proper parts. I will proceed to separate the two, and to examine them in turn.

2.1 Proposal 1: Proper Parts of Shapes

On the one hand, Armstrong’s account might be taken to suggest that shapes have parts, and that on the basis of the number of parts they have, a relation of part-to-whole can be established between them. So, for example, the triangle has three parts, each of which is a straight line, and the quadrilateral has four parts, each of which is similarly a straight line. Because the triangle has three straight lines and the quadrilateral has four straight lines, the triangle is a proper part of the quadrilateral.

For this proposal to work, however, all shapes must have parts of a single kind, for only then will it be their having a different number of parts that establishes a relation of part/ whole identity between them. Recall that in the case of the lengths, the property of being exactly two meters long is (supposedly) a proper part of the property of being exactly three meters long because the former is strictly identical with the property being exactly one meter long taken twice over while the latter is strictly identical with that same property taken three times over. Otherwise, one could not establish their part/ whole identity.

But it is not clear that in the case of shapes there is anything structurally like the parts of a length determinate. If Armstrong identifies the parts of shapes with their straight lines, as he explicitly does in the case of the triangle and the quadrilateral, then he faces the rather obvious difficulty that the proposal will not work in the case of shapes that have the same number of straight lines, such as the square and the rectangle. Nor will it work in the case of shapes that do not have straight lines at all, such as the circle, some irregular shapes and mere doodles. The bottom line here is that this proposal makes the unrealistic demand that all shapes be composed of different quantities of one and the same part.

2.2 Proposal 2: Properties of Proper Parts

On the other hand, Armstrong’s account might be taken to suggest that shapes are part/whole identical because they have determinate properties that are themselves part/whole identical. The thought would then be that shape X can be shown to be a proper part of shape Y, by showing that a proper part of X has a lesser quantity of a determinable D than the corresponding proper part of shape Y.

Armstrong mentions the properties of being straight and being a line. These properties are in fact unsuitable for establishing the part/whole identity of shapes, since they are not quantitative properties. Armstrong also mentions the property of having a particular length. And he could have added the property of having angles of a particular size, and having a particular surface area. These are more promising properties since, like the masses and the lengths, they might be taken to be quantitative in nature, in which case their different determinates might stand in the part/ whole relation to one another.

But it should be clear that this proposal is hopeless. Suppose that side a of shape X has a length of 10 cm, and side b of shape Y has a length of 20 cm. From the fact that the properties being exactly 10cm long and being exactly 20cm long are part/whole identical, it does not however follow that the things that have them, namely a and b, are themselves part/whole identical. Nor, therefore, does it follow that since a and b are proper parts of shapes X and Y, respectively, X and Y themselves are part/whole identical. It would otherwise be like saying that if my brain weighs 1.8 kg and my brother’s brain weighs 2 kg, my brain is a proper part of his brain. And further, that since my brain is a proper part of me and my brother’s brain is a proper part of my brother, I am a proper part of him!

This second proposal is further undermined by the fact that it trades on an ambiguity concerning the word ‘shape’. For by ‘shape’ one can either mean the property of having a shape, say the property of being a triangle or triangularity, or the figure which instantiates such a property. In short, ‘shape’ can either refer to a property (universal) or to an individual possessing that property (particular). Accordingly, the claim that shapes stand in the part/ whole relation can amount to either the claim that shape properties bear that relation, or to the claim that shaped figures do. Armstrong does not explicitly distinguish between the two claims. And while he clearly thinks of shapes as properties, his account shifts from focusing on features that belong to shape properties to features that belong to the figures that instantiate those properties. Armstrong talks about the lengths of the sides of the triangle. But this is a property of a triangular figure—it is the figure, and more specifically its sides that have the property of beingof a determinate length. The property of triangularity does not have the property of having sides of such-and-such a length. Otherwise, all triangular figures would have sides of the same length!

Given the failure of this second proposal, I conclude that there is no case here for thinking of shapes as being part/ whole identical. I think it is far more plausible to take Armstrong’s proposal to be that shapes are partially identical in the sense that they have parts in common, and hence in the sense of overlap rather than part/ whole. However, as we pointed out earlier, if shapes do not stand in the part/ whole relation, then the quasi-mereological account cannot explain the incompatibilities that hold between them. For only if shapes are related in this way can one explain their incompatibilities in terms of the identification of a whole with one of its proper parts. I conclude, therefore, that Armstrong’s quasi-mereological account simply does not have the resources to deal with the case of shape-incompatibility.

3 Is the Notion of A Structural Universal Coherent?

In the previous section, I argued against Armstrong’s quasi-mereological account of property incompatibility by showing how the account fails to accommodate certain cases of incompatibility among properties. In this section, I wish to argue against Armstrong’s account on more principled grounds, namely, that the notion of a structural property or universal which lies at the heart of that account is problematic. I will do so by considering an objection that David Lewis has raised against structural universals (1986a: 78–107). Lewis claims that structural universals involve the repetition of universals within their structure, but since universals are one, not many, this is impossible. Accordingly, there are no structural universals.9

Lewis asks us to consider the structural universal being methane. Necessarily, to be a methane molecule a particular must have (at the atomic level of analysis) one part which exemplifies the universal being carbon, and four parts each of which exemplifies the universal being hydrogen, with the carbon-part linked to the hydrogen-parts by the relation of bonding.

Thus, in the structural universal being methane, the universal being hydrogen and the relational property being a bonding occur not once, but four times over. Lewis objects that this is impossible, since universals are one, not many. Universals are postulated primarily to account for qualitative identity. Two apples can be the same shade of color, two tables can have the same shape, etc. The proponent of universals reduces such instances of qualitative identity to instances of numerical identity. When two objects are qualitatively identical with respect to color, shape, etc., there is some entity which they literally have in common–viz., a universal of color, shape, etc.10 If there could be several of a universal, however, then particulars could resemble one another by having ‘duplicates’ of that universal rather than by having the universal itself in common. And this would naturally undermine the explanation of qualitative identity in terms of numerical identity (as well generate a parallel problem about the qualitative sameness of the duplicates themselves).

Now, if universals are one, and not many, then what could it mean to say that the structural universal being methane contains four of the universal being hydrogen and four of the universal being a bonding? (Lewis 1986a: 91).11 Armstrong is aware of this difficulty, and by way of solution suggests that the universals that figure in a structural universal are of a particular sort, namely, particularizing universals. A particularizing universal is one where we can speak of an instance of that universal. For example, being one kilogram of lead is a particularizing universal, since we can speak of a kilogram of lead (whereas being red is not, since we cannot speak of a red). And, Armstrong writes,

there is a sense in which [particularizing] universals enfold particularity within themselves even when considered in abstraction from their instances. In the schematic example given of a structural universal—something of the F-sort having R to something else of the F-sort—F must be a particularizing universal: an F having R to another F. This is what permits repetition in the structure. It allows different non-relational elements in the structure to be different instances of the same universal (1986b: 88).

According to Armstrong, then, we can meaningfully talk of there being four instances of the universal being hydrogen in the structural universal being methane, because each instance is a particularizing universal. Particularizing universals are such that each of their instances is unambiguously one. They ‘enfold particularity within themselves’, and can thus occur repeatedly within a structural universal.

But I fail to see how the fact that each instance of the universal being hydrogen is unambiguously one instance shows that the universal being hydrogen itself can be counted several times over. Armstrong seems to get the desired conclusion by equivocating on the phrase ‘an instance of a universal’. In one sense of this phrase, an instance of a universal is simply a particular which instantiates that universal. An instance of the universal being hydrogen, for example, is a hydrogen atom. This sense does not, of course, license Armstrong to draw the conclusion that a universal itself can be counted several times over.

The sense that would license Armstrong to draw that conclusion is that according to which an instance of the universal being hydrogen is itself a universal—and the universal being hydrogen at that! But this is unintelligible. How can a universal have itself as one of its instances? And if it does, how could it have more than one? For there to be two (or more) instances of a universal, there has to be something in virtue of which the instances differ. However, there can be no mere numerical difference between universals such as there is between particulars. Universals can only differ qualitatively, i.e., in the features that define them. Thus for there to be two instances of a universal which are themselves that universal, for us to be able to distinguish between them, they would have to differ precisely in those features. But if they did, then they would no longer be two universals of a single universal (whatever that means!). As it stands, therefore, Armstrong’s notion of a particularizing universal is too obscure, and consequently his attempt to meet Lewis’ objection by appealing to that notion fails. Structural universals involving repetition of universals in their structure are not genuine universals.

Given the centrality of structural universals to Armstrong’s quasi-mereological account of property-incompatibility, it is worth considering whether there might be alternative ways of construing structural universals which avoid Lewis’ objection. One natural suggestion is to say that the structural property of (say) being methane is the property that any molecule has just in case it consists of five atoms, one having the property of being carbon and the other four having the property of being hydrogen, where each of the four hydrogen atoms stands in the appropriate bonding relation to the one carbon atom. Thus construed, being methane is a structural property insofar as only a suitably structured particular can instantiate it, viz., a methane molecule. Call this the Natural Construal of structural universals. The Natural Construal will still be unacceptable to Lewis, since he recognizes mereological fusion as the only mode of composition, and is accordingly blind to structure (Lewis 1986a: 36–8; 1991). But the claim that all composition is mereological is highly controversial, and can be put to one side here (but see Kalhat 2008b).

The Natural Construal of structural universals avoids postulating repeated universals within their structure. It does so, effectively, by shifting all structure away from the universals and on to the particulars that instantiate them. By doing so, however, the composition of the structural universals themselves becomes unclear. Should we think of being methane as composed of the universals having five hydrogen atoms, having one carbon atom and having each carbon-hydrogen pair bonded? These look too much like reified predicates. But even if we take them as genuine properties, their composition is now unclear. Intuitively, the universal having five hydrogen atoms is complex, composed of the universal having one hydrogen atom five times over. But we know that this cannot be so. So what is the composition having five hydrogen atoms? Are we not simply forced to conclude that the property is simple? The trouble with the Natural Construal of structural universals, then, is that it makes it unclear whether they have any composition at all. At the same time, Armstrong’s quasi-mereological account of property incompatibility requires that incompatible structural properties stand in the part/ whole relation to one another, and hence that they have composition, for otherwise the account will have no purchase on their incompatibility.

Armstrong could say that a particular that instantiates the property of being methane has a proper part that instantiates the property of being hydrogen, and yet maintain that the property of being methane is wholly distinct from the property of being hydrogen. This would still allow him to claim that to attribute two incompatible properties to one and the same particular involves identifying a whole (particular) with one of its proper parts. And he would no longer need to ensure that incompatible properties are themselves complex. But all of this would come at the cost of having to recognize a brute relation of necessitation holding between the properties of being methane and being hydrogen, since necessarily nothing can instantiate the one without instantiating the other. Armstrong’s actual account allows him to explain that relation in terms of the part/ whole relation (whether this explanation is genuinely reductive of necessity will be discussed in Sect. 4 below). But on the Natural Construal, the relation between the two properties is one of brute necessitation. This, of course, disrupts Armstrong’s combinatorial scheme, since ex hypothesi the two properties are wholly distinct, yet they do not have the unrestricted combinatorial freedom that Armstrong confers to wholly distinct entities (see Armstrong 1989: 69ff).

Bigelow and Pargetter offer a different way of construing the composition of structural universals, one which explicitly aims to avoid Lewis’ objection while still being capable of explaining the rich pattern of entailments that hold between a structural universal and its constituents (e.g., that if a molecule instantiates the structural universal of being methane, then necessarily parts of the molecule instantiate the universals of being hydrogen, being carbon, and bonding) (1990: 82–92). They develop a ‘three-level’ theory of universals, one which they hope will account for those entailments without making use of modal notions (or at least without making use of them in unacceptable, ‘black-magic’ sorts of ways). At level 1 we have individuals having various properties and standing in various relations to other individuals, e.g., methane molecules, hydrogen atoms, etc. At level 2 we have the properties and relations of these individuals, e.g., being hydrogen, bonding, etc. These include first-order properties and relations (i.e., ones not possessed in virtue of having other properties and relations), and also second-order properties and relations (i.e. ones which are possessed in virtue of having other properties and relations). Finally, at level 3 we have the relations or ‘proportions’ holding among the properties or relations of individuals. For example, the properties of being hydrogen and being part of this molecule and being carbon and being part of this molecule stand in ‘a relationship which is characterized by the proportion 4:1’ (1990: 87).

Given this framework, they propose to characterize the structural universal methane as a relational property of an object, or more particularly of a molecule. It relates the molecule to various properties. These properties are being carbon, being hydrogen, being bonded. Being methane, then, is to be identified with a highly conjunctive second-order relational property of an individual (molecule): the property of having a part which has the property of being hydrogen, and having a part which is distinct from the first part and which has the property of being hydrogen, and ... (1990: 87–8).

As we see here, Bigelow and Pargetter’s characterization of structural universals involves postulating universals of the form ‘having a part that Φ’s’, ‘having a part that is different from the first and Φ’s’, and so on. It is the bare numerical difference between the instances of these universals that is meant to distinguish between the universals themselves. But how is that done? How does the bare numerical difference of the instances register at the level of the universals themselves? Notice that unless they do, the properties of having a part that instantiate the property of being hydrogen, and having another part..., etc., all collapse into one and the same property, and then Bigelow and Pargetter’s theory of structural universals is also subject to Lewis’ objection. To ensure that the bare numerical difference between the hydrogen parts of the methane molecule register at the level of the universals they instantiate, non-qualitative properties of being those instances, i.e., hacceities, must be postulated. The bare numerical difference between the instances then registers at the level of the universals simply by having those haecceities among their constituents. Thus, the property of being a part that instantiates the property of being hydrogen is really the property of being a part that instantiates the property ofbeing that very partand also instantiates the property of being hydrogen; mutatis mutandis for the other properties.

Armstrong would be ill-advised to endorse Bigelow and Pargetter’s theory. Bigelow and Pargetter concede that the relation of necessitation holding between being methane and being hydrogen is not explainable in simple mereological terms (i.e., being hydrogen is not a proper part of being methane).12 They propose to explain the relation instead in terms of the proportions or level (3) relations previously postulated. They write:

The property of being methane clearly stands in a pattern of internal relations of proportion to other properties: being hydrogen, being bonded, being carbon, and so on. ... it is of the nature of methane that it stands in these relations to these properties. Standing in these relations is an essential characteristic of methane. These relations are then what we call internal relations. The related terms could not exist without standing in those relations. ... being methane is so related to being carbon that being methane cannot exist without standing in that relation to being carbon (1990: 88).

Bigelow and Pargetter thus explain the relation of necessitation holding between being methane and being hydrogen in terms of an internal relation, one which is itself grounded in the essence of being methane. They concede that this explanation still involves necessity. For one thing, the relation of necessitation is ultimately grounded in the notion of essence, which is arguably itself a modal notion. For another, the notion of an internal relation is modal in nature, since, as characterized by them, the relation is such that ‘the related terms could not exist without standing’ in it (italics added). Bigelow and Pargetter nevertheless claim that the ‘modal magic’ involved here is an instance of ‘white’ rather than ‘black’ magic (1990: 90); that is, it is a tolerable primitive necessity. However, the presence of any unreduced necessity in the relation between being methane and being hydrogen will disrupt Armstrong’s combinatorial scheme. For that necessity will rob the properties of the unrestricted combinatorial freedom which Armstrong grants to all wholly distinct properties.

But there is a second, more immediate problem with Bigelow and Pargetter’s proposal. Bigelow and Pargetter claim that being methane stands in relations of proportions to the properties of being hydrogen, being carbon, being bonded, etc. (1990: 88). What relations are these? Well, since there are four hydrogen parts in a methane molecule, being methane presumably stands to being hydrogen in the relation characterized by the proportion 1:4. (Strictly speaking, it stands in that relation to being hydrogen and being part of this molecule). But to say that being methane stands to being hydrogen in the proportion 1:4 is surely to say that there is one of the property of being methane to four of the property being hydrogen (if Bigelow and Pargetter meant that there is one instance of being methane to four instances of being hydrogen, then it would not be the properties that stand in the proportion 1:4 but rather the instances). I fail to see therefore how Bigelow and Pargetter’s theory of structural universals is any less susceptible to Lewis’ objection than Armstrong’s. Whether we take the relation between being methane and being hydrogen to be the part/ whole relation or a relation of proportion, the difficulty remains that we are committed to taking the universal being hydrogen as occurring four times over, which is impossible.

Bigelow and Pargetter’s theory of structural universals does not therefore give Armstrong the resources needed to meet Lewis’ objection. And if that objection cannot be met, then structural universals involving repetition of universals within their structure are not genuine universals. This conclusion also rules out as structural, all ‘quantitative’ universals such as being exactly five kilograms in mass and being exactly two meters long. For insofar as they can be taken to be composed of the universal being exactly one kilogram in mass five times over, and being exactly one meter long twice over, respectively, they obviously involve the repetition of universals within their structure.

Armstrong might try to avoid cashing out the structure of quantitative universals in such objectionable ways. For example, he might take being exactly five kilograms in mass to be composed of being exactly two kilograms in mass and being exactly three kilograms in mass instead. But this only pushes back the problem. For (say) being exactly three kilograms in mass can plausibly be taken to be composed of the universals being exactly two kilograms in mass and being exactly one kilogram in mass. Yet if so, being exactly five kilograms in mass turns out to be composed of being exactly one kilogram in mass and being exactly two kilograms in mass taken twice over, which is again problematic.

Armstrong might now try to settle for some other composition of being exactly three kilograms in mass, one that escapes Lewis’ objection. And of course, he will in a sense succeed in doing so, for quantitative universals such as being exactly three kilograms in mass can always be decomposed into constituent properties of unequal size. But the trouble with this strategy is that it either leads to paradoxical results or else is wholly arbitrary. When the universal being exactly three kilograms is a constituent of being exactly five kilograms in mass, it cannot be decomposed into the universals being exactly two kilograms in mass and being exactly one kilogram in mass, since the original universal will also have as a constituent the universal of being exactly two kilograms in mass, with the consequence that the latter occurs twice within it. On the other hand, when the universal being exactly three kilograms in mass is considered independently of the universal of being exactly five kilograms in mass, Armstrong would have no problem in decomposing it into the universals being exactly two kilograms in mass and being exactly one kilogram in mass (1997: 52–3). The upshot of this is that the universal being exactly three kilograms in mass does and does not have a given composition.

As a last resort, Armstrong might say that the situation is not in fact paradoxical, for the universal being exactly three kilograms in mass does and does not have a given composition in different contexts, viz., in the context of viewing it on its own and in the context of viewing it as a constituent of another universal. The situation would only be paradoxical if (say) when viewing the universal on its own, it both had and failed to have a given composition. However, to reply in this way is to avoid the charge of paradox only at the cost of a new charge of arbitrariness. For, surely, it should make no difference to the composition of a given universal, whether that universal is taken on its own or as a constituent of some other universal. One might even question whether it could really be the same universal, if it had one composition in one context and a different composition in another context. For having a particular composition is surely an intrinsic property of these universals. This defense, then, unacceptably turns the structure of structural universals into nothing more than a projection of an interest-relative line of analysis. I conclude, therefore, that Armstrong cannot meet Lewis’ objection, and consequently that he is forced to deny that quantitative universals such as being exactly three kilograms in mass and being exactly five kilograms in mass are structural universals.

Of course, if quantitative universals are not structural, then Armstrong’s quasi-mereological account cannot deal with the incompatibilities they give rise to. The quasi-mereological account requires not only that incompatible universals be partially identical, but also that they be structural. Mere partial identity will not do, since P&Q and P are partially identical, yet by definition, are instantiated by the very same particular, and are thus perfectly compatible. Thus, the rejection of structural universals involving the repetition of universals within their structure leaves Armstrong’s quasi-mereological account without the (non-modal) resources for dealing with a very large class of incompatible universals—indeed, the very universals that Armstrong thinks his account can take care of so well! (1997: 55).

Ultimately, however, what the rejection of quantitative universals as structural in nature shows is that these universals are not composed of further universals at all, and hence that they are simple. They cannot be structural for, as we have seen, they would then involve the illegitimate repetition of universals within their structure. So if they are complex, they must be conjunctive. But if they were conjunctive, then by definition their conjuncts would be instantiated by the very particulars that instantiate the universals as a whole. Yet obviously one and the same particular cannot instantiate (say) both being exactly three kilograms in mass and being exactly one kilogram in mass. Thus, quantitative universals must be simple. If so, not only does it follow that the quasi-mereological account cannot account for their incompatibilities. Since they are wholly distinct, Armstrong is also forced to recognize irreducible relations of incompatibility among wholly distinct entities–that is, necessity in re. This destroys the basic idea behind Armstrong’s combinatorial theory.

4 Does the Quasi-Mereological Account Presuppose Modality?

In the previous two sections, I have argued against the adequacy of Armstrong’s quasi-mereological account of property incompatibility. I now wish to challenge its reductive aspirations. It will be recalled that Armstrong develops the quasi-mereological account of property-incompatibility as a way of dealing with the fact that not all properties are compatible. Now, it is vital that the quasi-mereological account make no use of modal notions, for otherwise the combinatorial theory, of which the quasi-mereological account is a central part, will fail in its ultimate aim to reduce modality. On the face of it, it looks as if that account is indeed free of modal notions, for it explains property incompatibility (i.e., the impossibility of co-instantiation of certain properties) in terms of the part/ whole relation, itself a non-modal notion. In this section I argue, however, that far from being free of modal notions, the quasi-mereological account presupposes the modal notions of impossibility and necessity.

The core of Armstrong’s quasi-mereological account is the idea that to combine two incompatible properties is effectively to identify a whole with one of its proper parts. But, one ought to ask, what is the problem with this? Why cannot the whole be identical with one of its proper parts? Well, presumably because it is impossible for the whole to be identical with one of its proper parts. So far, then, the quasi-mereological account trades one impossibility, that of combining incompatible properties, with another impossibility, that of identifying the whole with one of its proper parts. Armstrong must now go on to offer a non-modal explanation of why it is impossible for the whole to be identical with one of its proper parts. Different answers to this question can be extracted, I believe, from Armstrong 1989 and 1997. Let us examine them in turn.

Armstrong 1989 writes,

It is plausible to say that the truth that a proper part of an entity is not identical with that entity is true solely by virtue of the meaning we attach to ‘proper part’ (1989: 80).

Armstrong here postulates analyticity as the source of the impossibility of identifying the whole with one of its proper parts. Appealing to analyticity weakens Armstrong’s combinatorialism, for analyticity is presumably a non-combinatorial source of modality (Bradley 1992: 217). Even so, it is unclear how exactly we are to construe Armstrong’s suggestion. How does the meaning of ‘proper part’ establish the impossibility of identifying an entity with one of its proper parts? There are, I think, two possible answers. First, one can say that the impossibility follows from the meaning of these words; secondly, one can say that the impossibility is partly constitutive of their meaning. Let us take these in turn.

The first suggestion, then, is that the impossibility of identifying the whole with one of its proper parts follows from the meaning of the expression ‘proper part’, much in the same way, I take it, that if a surface is red all over, then it follows that it is not green all over, even though that exclusion is not part of the meaning of the term ‘red’. Leaving aside the metaphorical and problematic notion of ‘something following from a meaning’, the problem with this suggestion is that the notion of ‘following’ employed here is modal in nature. For the sense in which the impossibility of identifying a whole with one of its proper parts is meant to follow from the meaning of ‘proper part’ is presumably that the meaning of ‘proper part’ entails that impossibility. But entailment is a modal notion: if P entails Q, then necessarily if P then Q. Thus, this first suggestion will not help Armstrong, for it makes vital use of the modal notion of necessity.13

The second suggestion is the Wittgensteinian one, according to which the impossibility of identifying the whole with one of its proper parts does not follow from the meaning of ‘proper part’, but is rather partly constitutive of that meaning. Such an impossibility is constitutive of the meaning of ‘proper part’ in the sense that to abandon it is effectively to change the meaning of ‘proper part’ (see Glock 1996: 240). Thus, if someone were to insist that a whole can be identical with one of its proper parts, then that person would have given a different meaning to the expression ‘proper part’ (or else, we would conclude that that person does not understand what ‘proper part’ means). I think this second suggestion is more promising than the first, for it offers a more concrete picture of what the link might be between meaning and modal notions such as impossibility and necessity. However, it is not a suggestion that Armstrong can endorse. For insofar as it takes the impossibility of identifying the whole with one of its proper parts to be, at least in part, what the expression ‘proper part’ means, then far from reducing it, it has placed that modal notion right at the heart of it. It is analytic that a whole cannot be identical with one of its proper parts because the impossibility of identifying the whole with one of its proper parts is partly constitutive of the meaning of ‘proper part’. I conclude, therefore, that Armstrong’s attempt to reductively explain the impossibility of identifying a whole with one of its proper parts in terms of analyticity fails.

Armstrong 1997, though still endorsing analyticity, points to a different way of explaining what the problem is with identifying the whole with one of its proper parts. The problem is that to do so is to render the whole non-identical with itself—the implication being that this is absurd (1997: 54). But why is it absurd? Armstrong does not say, but presumably he takes it as read that everything is necessarily identical with itself. I do not wish to dispute the assumption, but only to point out that it undermines Armstrong’s suggestion. The force of Armstrong’s suggestion depends on self-identity being necessary. For if a whole is only contingently self-identical, then it only contingently fails to be identical with one of its proper parts (or with anything else for that matter). And the contingent non-identity of a whole with one of its proper parts does not then explain the impossibility of an object instantiating incompatible properties—it only explains why no object in fact instantiates such properties. So, it is essential to Armstrong’s suggestion that it be impossible for an object to be non-identical with itself. But now, to explain the impossibility of a whole being identical with one of its parts in terms of the impossibility of a whole failing to be identical with itself is evidently not to offer a reductive explanation.

Armstrong need not stop here, however.14 Armstrong 1997 attempts to provide truthmakers for modal truths. A truthmaker for a true proposition P is that portion of the world in virtue of which P is true (1997: 13; 2004). Armstrong suggests that the truthmaker for a modal truth ‘will make that truth true in virtue of nothing more than the relations of identity... and difference holding between the constituents of the truthmaker’ (1997: 150). These relations, Armstrong goes on to say, are internal relations. Internal relations, as we have already seen, are necessitated by their relata, and, in accordance with Armstrong’s doctrine of the ‘ontological free lunch’, are no ‘addition of being’ to the relata themselves (1997: 12–13). The upshot of these claims, the story goes, is that the truthmaker for the necessary proposition ‘a is identical with itself’ is ultimately nothing but a itself. And on the assumption that a is non-modal in nature, then it appears that Armstrong can provide a genuine reduction of the necessity of self-identity after all, and hence of the impossibility of identifying a whole with one of its proper parts.

I disagree. There are a number of difficulties with Armstrong’s proposal here, not least with the idea that internal relations are ‘no addition of being’. While such relations are necessitated by their relata, they are not identical with them. a, an object, is not the same as the relation that holds between a and itself. If internal relations exist and are not identical with their relata, then they must be additional entities to the relata. Pace Armstrong, then, internal relations are additions of being. They are further items on the inventory of what there is.

Now, if the relation of self-identity is an additional entity to a, then it is wrong to posit merely a as the source of the necessity of a’s identity with itself. The source is something more specific: it is the necessity that attaches to the relation of identity in which a stands to itself. In other words: the source of the necessity of a’s identity with itself is the necessity of a’s identity with itself. Hardly a reduction of necessity! The point can also be made in terms of truthmakers, Armstrong’s preferred idiom. A truthmaker is that in virtue of which a truth is true. The necessary truth ‘a is identical with itself’ is not true, however, merely in virtue of a. It is true in virtue of something specific about a, namely, the modal character of the relation of identity in which a stands to itself, i.e., the necessity of its self-identity. To think otherwise is to confuse the relation of truthmaking with mere necessitation (a by itself, of course, necessitates the truth of ‘a is identical with itself’, since it is impossible for a to exist and for ‘a is identical with itself’ to be fail to be true. See Kalhat 2008a for more detail). The question as to the source of the necessity of self-identity remains, and to that extent, Armstrong must take the necessity as primitive. His attempt to provide truthmakers for modal truths does not help, therefore, in reducing the necessity implicit in self-identity.15 I conclude that Armstrong’s quasi-mereological account presupposes modality in a way that undermines the reductive aspirations of the combinatorial theory of which it is a central part.

If the objections raised in Sects. 24 are correct, then they show that Armstrong’s quasi-mereological account of property incompatibility fails. Without this account, however, the combinatorial theory has no non-modal resources for dealing with the array of impossibilities that the unrestricted combination of properties gives rise to. The combinatorial theory is thus forced to take at face value the relations of compatibility and incompatibility holding among properties in order to restrict the combinations to only those that constitute genuine possibilities. These relations, however, are modal in nature, for two properties are compatible if it is possible to combine them together and incompatible if it is impossible to do so. Thus, while recognizing these relations saves combinatorialism from generating all manner of impossibilities as well as possibilities, doing so undermines the promise of providing a reduction of the notion of possibility. Armstrong suggests that were his combinatorialism to turn out to be circular, he would adopt the fallback position of viewing it, not as a reductive analysis of possibility (and necessity), but as an ‘attempt to exhibit in a perspicuous manner the structure of modality’ (1989: 139). If my arguments in this paper are correct, then this is the best he can hope for.16

Footnotes
1

It is unclear whether Armstrong still subscribes to combinatorialism. In his recent book Truth and Truthmakers, Armstrong argues that the relation between a particular and the universals it instantiates is necessary. This robs particulars and universals of the combinatorial freedom they require to yield all the possibilities (see Armstrong 2004 and Kalhat 2008a).

 
2

There are also incompatibilities (and entailments) among relations, but I shall not dwell on those here. For Armstrong’s handling of incompatibilities among relations, see Armstrong 1989: 84–6.

 
3

Armstrong does not think that the relation between a complex property and its constituents is merely the part/ whole relation. For example, the conjunctive property P&Q is something more than the mereological sum P+Q-the ‘something more’ being the requirement that the same particular which instantiates P also instantiates Q. But having said this, Armstrong thinks that ‘this extra condition does not abrogate the simple mereological relations that hold between [them]’ (1997: 53). So, while Armstrong’s conception of properties is only ‘quasi-mereological’, I will nevertheless follow him in speaking of the relation between a complex property and its constituents as the part/ whole relation.

 
4

A more straightforward explanation for the incompatibility of the properties being exactly two kilograms in mass and being exactly five kilograms in mass is that since the former is the property of being at least two kilograms and not more than two kilograms in mass and the latter is the property of being at least five kilograms and not more than five kilograms in mass, the incompatibility is an outright logical contradiction. But not all property-incompatibilities reduce in this way to logical contradictions, e.g., shape incompatibilities (see below for further examples).

 
5

For his naturalism, see Armstrong 1997: 5–6.

 
6

Armstrong gives an account of the resemblance of determinates of a determinable in terms of partial identity (1997: 55; 1978: 116–124). Determinates of a determinable resemble one another (to varying degrees) because they are partially identical. The closer the resemblance between any two determinates, the more they overlap, and hence the closer they get to complete identity.

 
7

Armstrong is aware of this difficulty; he writes: ‘It may be conceded at the outset that the unity of the class of shapes is a much messier affair than lengths, durations, masses which all arrange themselves as simple one-dimensional arrays’ (1997: 55).

 
8

This is not to say that all complex properties are conjunctive properties. A complex property is conjunctive only if it is the very same particular that instantiates each of the conjuncts; otherwise, it is structural.

 
9

Lewis also objects to structural universals on the grounds that they violate the Principle of Uniqueness of Composition, which says that for a given number of parts, there is only one whole that they compose. Kalhat 2008b refutes this objection.

 
10

This is, in essence, Armstrong’s ‘One over Many’ argument for universals. He writes: ‘I would wish to start [by saying that] many different particulars can all have what appears to be the same nature and draw the conclusion that, as a result, there is a prima facie case for postulating universals’ (1980: 102).

 
11

Lewis’ objection does not pose a similar problem for conjunctive universals. Since the very same particular which exemplifies a conjunctive universal also exemplifies its conjuncts, the situation where that particular exemplifies a universal F, and exemplifies another F, and so on, simply cannot arise.

 
12

It might be objected that on Bigelow and Pargetter’s account, being hydrogen is a part of being methane, since it is a part of the property of having a part that instantiates being hydrogen, and the latter property is indeed a part of being methane. However, if being hydrogen is taken to be a part of the property of having a part that instantiates being hydrogen, then being hydrogen will also be a part of the property of having a part that is distinct from the first and instantiates being hydrogen, and having another such part, etc. But then the property of being hydrogen occurs four times over in the structural universal being methane, which is impossible. Bigelow and Pargetter must therefore deny that being hydrogen is a part of the properties of having a part that instantiates being hydrogen, and having another such part, etc. This means that they must either deny the transitivity of the parthood relation, or deny that the more complex property of having a part that instantiates being hydrogen is a part of being methane at all.

 
13

Couldn’t Armstrong understand ‘following from’ syntactically, in terms of derivability in some rich logical system? The matter is complex and cannot be pursued here, but let me indicate why I think that this possibility is not open to Armstrong. The notion of derivability itself appears to hide modality, i.e., ‘X is derivable from Y’ means ‘it is possible to derive X from Y’. To avoid the modal construal of derivability, the combinatorialist should, I think, insist that ‘X is derivable from Y’ means that there is a derivation of X from Y. But since, of course, not all derivations have in fact been carried out, this suggestion requires a commitment to Platonism: derivations are abstract objects. A naturalist like Armstrong, however, cannot countenance abstracta. Therefore, I do not think that he can pursue the syntactic approach, in which case he must embrace the standard and modal understanding of ‘follows from’ as entailment. (See Lewis 1986b: 150–57 for related difficulties).

 
14

I thank the referee for this journal for the suggestion that follows.

 
15

For what it is worth, I do not think that that necessity can in fact be reduced.

 
16

Thomas 1996 and Hippaka et al. 1999 defend the viability of non-reductive versions of combinatorialism.

 

Acknowledgments

I wish to thank Hanjo Glock for valuable comments on previous drafts of this paper.

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© Springer Science+Business Media B.V. 2010