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In defence of structuraluniversalsD.M. Armstrong aa University of Sydney

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To cite this article: D.M. Armstrong (1986): In defence of structural universals,Australasian Journal of Philosophy, 64:1, 85-88

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Australasian Journal of Philosophy Vol. 64, No. 1; March 1986

DISCUSSION

IN DEFENCE OF STRUCTURAL UNIVERSALS

D. M. Armstrong

1. The central issue. At the heart of David Lewis' case against structural universals lies his contention that two different things cannot be composed of exactly the same parts.

Here is what I take to be a counter-example to his principle. Let a and b be two particulars, and R be a non-symmetrical relation. Let it be the case that a has R to b, and that b has R to a. We have two distinct states of affairs ('two different things'), yet, in a clear sense of the word 'composed', they are composed of exactly the same parts: a,b and R.

The two states of affairs may be called structures. In his important recent book The Categorical Structure o f the World (1983, Section 101), Reinhardt Grossman offers the following identity-conditions for structures. S~ and $2 are the very same structure if and only if (a) they contain the very same non- relational parts; (b) they contain the very same relations; (c) the same parts stand in the same relations to each other. In my counter-example, the two structures contain the very same non-relational parts, the very same relations, but it is not the case that the same parts stand in the same relation to each other.

My counter-example to Lewis' principle was chosen because, although it involves structures, it does not involve structural universals. This shows, I think, that the difficulty raised by Lewis is best thought of as an argument against postulating any universals, structural or otherwise, or, at least, as an argument against postulating relations which are universals.

Lewis, of course, would not allow the counter-example. By far the simplest way for him to deal with it is by adopting a philosophy of what, following D. C. Williams (1953), and, more recently, K. K. Campbell (1981), he calls 'tropes'. Tropes are properties and relations, but they are properties and relations conceived not as universals but as particulars. On this view of relations, my alleged counter-example becomes two states of affairs, a R1 b, and b R2 a, where R~ and R2 are not identical, although they may resemble exactly. (The universal R perhaps reduces to an equivalence-class of exactly resembling tropes.) Given this account, I have certainly not produced a counter-example to Lewis' view that two different things cannot be composed of exactly the same things.

But is not the dispute now a stand-off? Lewis can use his principle against a philosophy of universals. I can use universals to produce a counter-example to his principle. Indeed, is not Lewis close to begging the question against me?

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86 In Defence of Structural Universals

It may be replied that Lewis' view is the more economical. He puts forward an attractive-sounding principle. I have to deny that the principle holds in all cases, and my reason is that it is defeated by those suspicious characters: universals.

To this I reply that economy in a metaphysics can only be judged, as Mark Johnston has put it to me, 'in the end-game'. For myself, I believe that universals are great explainers. The loss on the roundabouts as a result of having to deny Lewis' principle may well be made up with interest on the swings. In any case, as the great Dr. Tarrasch said, 'before the end-game, the Gods have placed the middle-game'.

What it would be nice to have, but what I cannot supply, is formal description of an operation which will take one from any unordered set of universals to possible structural universals which involve nothing but members of the set. (I say 'possible' in order to respect the Principle of Instantiation which I believe should apply to all universals.) Such an operation will permit the one universal in the original set to appear in more than one 'place' in the structural universal. (E.g. an F having R to an F which has R to a third F.) A parallel is the way that, in a set of sets, the very same individual may be found as a member of different sub-sets.

2. Tropes. Lewis agrees with me that we need an objective distinction between natural and unnatural classes. In a previous paper (1983) he remained neutral between three ways of accounting for the distinction: holding a sparse theory of universals, taking the naturalness of certain classes as primitive, or using a complex primitive notion of similarity. Now he says that he has turned against universals for the reasons given in his paper, but that he should have mentioned, as another alternative, a sparse theory of tropes.

I am inclined to think that, for accounting for the natural/unnatural class distinction, by far the best prospects are universals and tropes. The alternative theories seem to face very serious difficulties with regard to relations.

A universals theory takes relations to be entities: they are types. A trope theory takes relations to be entities: they are tokens. But if universals and tropes are denied, then relations have to be constructed out of pairs (more generally, out of n-tuples) of particulars.

These pairs, however, will have to be ordered, rather than unordered, pairs. That this is so can be seen by considering our non-symmetrical relation R. We must either postulate a primitively natural class of pairs, or else appeal to a complicated primitive similarity holding between each member of the class of pairs. But what of the case where the relation holds in both directions: a R b and b R a? If we simply use unordered pairs, then we cannot distinguish, as we want to do, between the two states of affairs. With ordered pairs, however, we can have both and .

But what of the notion of order? It cannot be explicated in terms of relations. It will have to be taken as a primitive predicate, or rather as a series of primitive predicates corresponding to the polyadicity of the relation in

i John Bigelow and Peter Forrest have independently given some preliminary thought to this question.

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D. M. Armstrong 87

question. That seems to be an unwelcome addition to the apparatus required by the Natural class and the Resemblance theory. (The Universals theory and the Trope theory, by contrast, can rather naturally treat the polyadicity of their entities as properties of these entities, differing only as to whether these second-order properties are universals or particulars.)

Can we use the Wiener-Kuratowski device, and substitute for the ordered pairs an unordered set of sets? For we substitute, perhaps, { [a}, [a,b}}, and for {{b}, [a,b}. However, as a piece of serious metaphysics, this seems quite unacceptable. For a start, the correlation between ordered pairs and unordered sets of sets is quite arbitrary. The substitution just given could as well have been reversed. Still worse, the existence of unordered sets follows from the mere existence of the particulars a and b; but, in general, a R b does not so follow. (I owe this point to Paul Hager.)

So a Natural class view, or a Resemblance view, must treat the notion of an ordered n-tuple as primitive, yet cannot take this ordering to be a primitive relation. This is conspicuously messier than treating relations as particulars, and only afterwards inquiring what it is that unifies certain classes of such particulars.

A more familiar difficulty for the traditional Natural class view is the possibility that two distinct properties or relations might be co-extensive. (A similar difficulty arises for the traditional Resemblance view.) There would then be two properties (or relations), but only one primitively natural class. This difficulty, however, does not directly affect Lewis, because his natural classes span different possible worlds. Given the plausible assumption that distinct properties and relations are separable, the classes associated with distinct properties and relations will always be distinct, though overlapping, classes.

Lewis' extra freedom here is ambiguous, however. For him, it shows part of the power of a Realistic view of possible worlds that it solves the co- extension difficulty so simply. But suppose, as I would suppose, that such an anti-Naturalist view of possibility is to be avoided if at all possible. Then we have a further argument in favour of the tropes. For with the trope theory two co-extensive properties or relations fall into two wholly disjoint classes of tropes, even in this world.

Once again I think the moral is: universals or tropes.

3. Particularising universals. As between these two contestants, I, of course, would choose universals. Suppose that F is a monadic universal, and R a dyadic one. We then have a possible structural universal: an F having R to another F. If there actually are two F-particulars related by R, then, I would say, we have an actual structural universal. This structure has two marks which, I suggest, are jointly sufficient to make the structure a universal. First, it is identical in its different instances. Second, it is indefinitely repeatable (any place, any time).

I would like to put forward a hypothesis about the nOn-relational components of a structural universal: in this case the universal F. I think that they must be what I have elsewhere called a particularising universal

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88 In Defence o f Structural Universals

(1978, Chapter 11, Sec. IV). A particularising universal is one where we can speak of an instance the universal. Suppose that greenness is a universal. It is not a particularising universal, because we cannot speak of a green. Suppose that being lead is a universal, but that the atomic theory of lead is false and lead is everywhere homogeneous. We could not speak of a lead, and lead would not be a particularising universal. (In Wilfrid Sellar's terms, greenness and being lead would both 'lack grain'.) Contrast an electron, a circular green pattern of size S, a kilogram of lead.

The distinction links up with, but does not exactly parallel, Quine's distinction between those referring phrases which do, and those which do not, 'divide their reference'. The parallel is not exact because while 'an electron' divides its reference, 'a kilogram of lead' does not. Consider how many different particulars, each of them a kilogram of lead, there are in a 2-kilogram lump of lead.

We can, of course, speak of a green thing or a lead thing. But here I think that if what are involved are really to be universals, then the word 'thing' will have to be a place-holder for something more determinate. In the case of 'green thing' we might perhaps have 'green circular patch of size S'. The latter predicate picks out a definite particular with definite boundaries. So it is at least not implausible to think that a particularising universal corresponds to the predicate.

The justification of the phrase 'particularising universal' should now be apparent. There is a sense in which such universals enfold particularity within themselves even when considered in abstraction f rom their instances. In the schematic example given of a structural unversal -something of the F-sort having R to something else of the F -sor t -F must be a particularising 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.

I should like to acknowledge the assistance of Mark Johnston and David Lewis in composing this reply.

University o f Sydney Received January 1985

REFERENCES Armstrong, D. M. (1978) Universals and Scientific Realism, Cambridge University Press. Campbell, K. K. (1981) 'The Metaphysics of Abstract Particulars', Mid-West Studies in

Philosophy, 6. Grossman, R. (1983) The Categorical Structure of the World, Indiana University Press. Lewis, D. K. (1986) 'Against Structural Universals', Australasian Journal of Philosophy 64,

pp. 25-46.

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