Are Subclasses Parts of Classes?

Are Subclasses Parts of Classes?Author(s): Alex OliverSource: Analysis, Vol. 54, No. 4 (Oct., 1994), pp. 215-223Published by: Oxford University Press on behalf of The Analysis CommitteeStable URL: http://www.jstor.org/stable/3328808 .

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Are Subclasses Parts of Classes?

ALEX OLIVER

In Parts of Classes [5], David Lewis maintains:

(1) x is a member of a set S iff the singleton of x is a mereological part of S.

But why should we accept this?1 It is uncontroversial that

(2) x is a member of a set S iff the singleton of x is a subset of S.

It is controversial whether the right hand side of (2) is conceptually prior to the left hand side, and thus whether it can be used to explain the mean-

ing of the left hand side. For the standard way of explaining the subset relation is via a definition in terms of membership:

(3) a set S is a subset of a set T iff every member of S is a member of T.

But let us not pause to worry about this admittedly obscure matter of conceptual priority. The focus of this paper is the move from (2) to (1). This cannot be characterized as the simple identification of the subset relation with the mereological part-whole relation restricted to the field of sets. For Lewis claims that, of the subsets of a given set, only the non-empty subsets of the set are parts of it. The empty set, a subset of every set, is not a part of any set. Along with this restriction goes a change of terminology. This change is subversive, being propaganda for Lewis's crucial claim that the empty set and the membered sets are different sorts of things, belonging to distinct ontological categories. But I will sometimes adopt his terminology for ease of quotation. Lewis calls those things which have members 'classes', those things which are members but which do not have members, 'individuals'. Hence, you, me and the empty set are individuals, and among the membered classes are both the old non-empty sets and the proper classes, those things which have members but which, unlike sets, are not members. The field of the subclass relation is restricted to classes and obeys the following equivalence:

(4) a class S is a subclass of a class T iff every member of S is a member of T.

Therefore, the empty set, being an individual, is not a subclass of any class.

1I accepted (1) for the purposes of argument in [6]. There my aim was to show that two Australian attempts to dissolve Lewis's mystery of singletons, taking (1) as given, did not succeed on their own terms.

ANALYSIS 54.4, October 1994, pp. 215-223. ? Alex Oliver



z16 ALEX OLIVER

In fact, Lewis also claims that the only parts of a class are its subclasses. So the Main Thesis of his Parts of Classes is:

(MT) The parts of a class are all and only its subclasses.

I am not concerned to argue that a class has parts in addition to its subclasses. Instead, I want to question Lewis's argument for one of the premisses of the Main Thesis, his First Thesis:

(FT) One class is a part of another iff the first is a subclass of the second.

Now Lewis thinks that his First Thesis is so evident that it needs no argument, unlike his Second Thesis that no class has any part that is not a class. But, quite rightly, he does not seem to think that his claim of self- evidence will convince. So having set down his First Thesis, he asks 'what can I say in its favour?'. I suppose that considerations in favour of a thesis are reasons to believe it, and so I do think there is something like an argument, or arguments, for his First Thesis.

1. Lewis gives three reasons, all of which I find unpersuasive. First, he claims that 'it conforms to common speech'. He cites three pieces of evidence for this conformity:

(a) that it is natural to 'say that a subclass is part of a class: the class of women is part of the class of human beings, the class of even numbers is part of the class of natural numbers, and so on.' ([5], pp. 4-5)

(b) 'it seems natural to say that a hyperbola has two separate parts - and not to take that back when we go on to say that the hyperbola is a class of x-y pairs.' (p. 5)

(c) in German, 'a standard word for "subset" is "Teilmenge", literally "part-set".' (p. 5)

He then proposes two explanations of this conformity to common speech, a devious and a straightforward explanation, the latter of which is to be preferred. The devious explanation is that our talk is mere metaphor, shored up by a formal analogy between the subclass relation and the mereological part-whole relation. The straightforward explanation is that we know that subclasses are parts of classes and 'we speak accordingly'([5], p. 5).

We have to pick apart Lewis's pieces of evidence one by one. The evidence cited in (a) is very weak indeed. I think Lewis is just plain wrong and Bostock is plain right when he says 'we do not very naturally say that one set is part of another set at all' ([3], p. 125). We do not say that the class of women is part of the class of human beings, though we do say that women are part of the human race. Lewis would have been on better ground had



ARE SUBCLASSES PARTS OF CLASSES? 217

he said that 'it is natural to say that a subclass is included in a class'. Evidence (b) is cheating. Of course, it is natural to say that a hyperbola

has two separate parts if by 'hyperbola' one means a (necessarily partial) token hyperbola drawn on the page. For then the hyperbola can be divided into two spatial parts. We also talk of types of geometrical objects as having parts, so we can say that a type of a particular hyperbola has two separate parts. But I would construe this talk as mere metaphor: the type has metaphorical parts because any token of the type has real spatial parts. Presumably, Lewis is talking about types, not tokens, since he wants to identify the hyperbola with a class of x-y pairs. Since I do not believe that our talk of parts of geometrical types is more than a metaphor, I question Lewis's evidence at the first step.

But let us give Lewis a run for his money and suppose that geometrical types really have parts. Then I think we should criticize Lewis's uncritical set-theoretical reductionism. If we are both thoughtful and mathematically knowledgeable, we should not put any weight on the claim that a hyperbola 'is a class of x-y pairs'. This is not just for Benacerrafian reasons involving the variety of representations of points (which origin? which axes? why not polar co-ordinates? why not triples, rather than pairs, given that hyperbolas are sections of three-dimensional figures?). There is also a variety of representations which need not involve either points or classes. Namely, a hyperbola is (i) a class of points; (ii) a class of lines (its tangents); (iii) a quadratic equation, of which Lewis's x-y pairs are the solutions. In the face of these equally valid representations of a hyperbola, we do not have to take back our spatial language about the parts of a hyperbola. The so-called 'identifications' of the hyperbola are not genuine identifications with any one of the representations, but are rather reasoned substitutions. The point has nothing particularly to do with the spatial language of parts applied to a hyperbola. For example, an ellipse is finite, a hyperbola is not, even though Lewis would identify both with an infinite class of pairs.

Notice how selective Lewis is in his choice of example. Consider Armstrong's list of cases of the part-whole relation (we should look at Armstrong because he is one of Lewis's sources for part-whole intuitions) ([1], pp. 36-7). It includes the following two cases: the proposition p is a part of the proposition p&q; the property P is a part of the property P&Q. Now accept Lewis's set-theoretical identifications of propositions and properties, as sets of possible worlds and sets of actual and possible individuals, respectively. It turns out that we cannot now preserve our talk of parts and wholes once the identification with sets has been made, if Lewis is right that the non-empty subsets of a set are its parts. For example, the set identified with the property P has the set identified with the property P&Q as a subset, not vice versa (similarly, for propositions). Do not think



218 ALEX OLIVER

that Lewis could simply disagree with Armstrong on these cases

(although this would still be picking and choosing if it were to work). At least in the case of properties, he is committed to Armstrong's position. His example: 'God's foreknowledge is said to be part of His omnis- cience'([5], p. 75).

As a final decisive point, it should be said that if Lewis's argument works at all it cuts both ways. We do not think it right to say that the number one is a part of the number two, and we do not think that we should take this back when we go on to identify (according to von Neumann's plan) one with {0) and two with {0, {0}}, where 0 is the empty set.

Lewis's evidence from German is hardly good evidence. It is true that

'teil-menge' is literally 'part-set' but it means 'subset'. Indeed, the emphasis should be on 'subset', for it does not mean 'subclass' (i.e. nonempty subset). Lewis's part-whole thesis does not work for subsets. Suppose that, for example, the subsets of {a), are all and only its parts. Then the parts of

{a) are {a) and 0. But how can this be? If {a) has 0 as its only proper part, then how does (a) differ from {b), since 0 is its only proper part too?

Lewis ought also to have pointed out that Bolzano used 'Teil' for members of sets, not subsets ([2], ??82-4). And, of course, in English we ordinarily talk of sets 'containing' their 'elements' or 'members' (my dictionary: member = part or organ of body). And it is just as well that Lewis does not know any French for the French for set is 'ensemble', literally 'whole'; and the French for subset is 'partie', literally 'part'. But the parts are not parts of the whole. It is the members of the set of which the set is the 'ensemble', and so the 'ensemble des parties' is not the original set, but the power set. Lewis must pick and choose his linguistic evidence, ignoring all these

examples because they contradict his claim that it is only the subclasses that are parts of a class, not the members and the empty set as well.2

In summary, the devious explanation of Lewis's evidence seems to be the

right one. It may be more complicated than Lewis's straightforward explanation, but that is because his linguistic evidence is a rag-bag masquerading as a silk purse.

2. Now let us consider what else Lewis says in favour of his First Thesis. His second point is that between the mereological part-whole relation and the subclass relation 'we have at the very least an analogy of formal char-

acter, wherefore we are free to claim that there is more than a mere

2 Notice that Lewis could not say that the parts of a class are all and only its subclasses, members and the empty set. Consider the two classes, {{a)) and (a). The parts of ((a)) would be {{a)), {a), and 0, and the parts of {a) would be {a), a and 0. So both classes can be divided exhaustively into the same proper parts, namely, a and 0. Then the

question is: how do these two classes differ?



ARE SUBCLASSES PARTS OF CLASSES? Z19

analogy' ([5], p. 5). The analogy of formal character is, as Lewis recog- nizes, really a necessary condition for his First Thesis rather than a reason for it. Not just any structure which satisfies the axioms of mereology will count as containing a part-whole relation. A structure which satisfies the mereological axioms is simply a complete Boolean algebra minus the null element. There are obvious examples of such a structure which we would not count as containing a part-whole relation. Think of a diagram which represents a structure with a genuine part-whole relation. Labelled dots represent the objects in the domain and an arrow joins one dot to another just in case the object represented by the first is a part of the object represented by the second. Of course, the diagram can be seen to satisfy the mereological axioms: the domain contains the dots and the partial order- ing is the relation of being joined by an arrow. But we would not say that one dot is part of another.

Here is another more specific and amusing example. Take any finite set of prime numbers, together with their product and their various partial products. Example: {2, 3, 5, 6, 10, 15, 30). This is a Boolean algebra (8- element, in this case) minus its null element 1. So we can define 'x is a part of y' as 'x divides y', and the fusion of any of the relevant elements as their lowest common multiple. Yet obviously we would not want to say that one number is a part of another just because the first divides the second, notwithstanding the fact that the German for 'divisor', 'divide' etc. is 'Teiler', 'teilen' etc.!

I also think there is a quite straightforward explanation of why the analogy of formal character exists which does not take the analogy to be more than analogy. The formal analogy between the mereological part-whole relation and the subclass relation was recognized very early on in the history of formal theories of parts and wholes (see, for example, [4], [8], [9]). That the analogy exists is no accident. If one is aiming to construct a formal system which describes the properties of a part-whole relation one will naturally seize upon a structure which has already been described, is simple and elegant, and which has enough of the right properties to serve as a structure containing the part-whole relation. The Boolean algebra of sets, developed and refined in the late nineteenth century, was precisely such a structure. One only has to make the generalizing move of viewing the axioms of the Boolean algebra of sets as susceptible of different inter- pretations; on the one hand, sets and the subset relation, and, on the other hand, parts and wholes and the part-whole relation. In the latter case, of course, the null element has to be deleted. From the point of view of the characterization of a Boolean structure as based on the operations of union, intersection, complement and inclusion and their interrelations, dropping the null element is a rather severe change because half the struc-



22zzo0 ALEX OLIVER

ture goes with its deletion. But if we think of a Boolean algebra pictorially, then dropping the null element can easily be seen to be an option and one which it is very natural to take.

Lewis sees the formal analogy as no accident because it is no mere anal-

ogy. I say the analogy is no accident, but still only an analogy. One should also remember that mereology, as it is presented by Leonard and Good- man, is a calculus of individuals. In other words, while they explicitly acknowledged the formal analogy between their calculus and the Boolean

algebra of sets, they did not think that it was natural to call the non-empty subset relation holding between sets, a part-whole relation.

3. Lewis's third and final consideration in favour of his compositional thesis is that the thesis is fruitful. The sort of fruitfulness he has in mind is

epistemological: the claim helps us understand the objects of set theory and the relations which hold between them. Dissecting his definition of

membership given in (1), we see that Lewis has two primitive relations; the relation (in fact, function) which holds between a thing and its singleton (the singleton function) and the part-whole relation holding between classes (the part-whole relation is used to define the fusion operation; union of classes is a special case of fusion). Lewis claims that by dividing the generative operations of set theory into two he has isolated the 'innocent Dr Jekyll' (fusion of classes) from 'the extravagant and powerful Mr

Hyde' (the singleton function). Thus the singleton function is blamed for the 'mysterious' hierarchy of sets which can be built up from a single indi-

vidual, and for the restrictions needed to avoid paradox. In contrast, the

mereological structure of classes is, says Lewis, ontologically innocent and well-understood. So, given the Main Thesis, at least one aspect of set

theory is made less mysterious. I am not convinced that Lewis's theory is fruitful in the way he suggests.

First, consider the following passage:

We understand how bigger classes are composed of their singleton atoms. That's the easy part: just mereology. That's where we get the

many into one, the combining or collecting or gathering. Those intro-

ductory remarks ... introduced us only to the mereology in set theory. ([5], p. 31)

The introductory remarks to which he alludes are a collection of quota- tions from texts on set theory such as 'a set is a collection of objects ... [It] is formed by gathering together certain objects to form a single object.' ([7], p. 238) Lewis thinks that he has explained the truth of these remarks. But he has not. The objects referred to in these remarks are not the singletons of the members of a set, but the members themselves. It is only by



ARE SUBCLASSES PARTS OF CLASSES? ZZr

interpreting these remarks in an unintended way that Lewis can claim that they are true.

Do 'we understand how bigger classes are composed of their singleton atoms'? It all depends on what Lewis means by 'understand'. If he means that we know the formal properties of the set-theoretic union operation, then I agree. But his identification of this operation with the mereological fusion operation adds nothing to this understanding.

What more do we understand through Lewis's identification? Lewis claims mereology is 'ontologically innocent' and his idea seems to be that this innocence should make the composition of classes from their singleton subclasses transparent to the understanding. But 'ontological innocence', as Lewis explains it, is not well-understood. Here is how he explains the phrase applied to mereology:

... we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many. ([5], p. 87)

Or, again, ... given a prior commitment to cats, say, a commitment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it. (p. 81)

I find it hard to understand these passages. To begin with the sentences 'it just is them', 'they just are it', 'this one thing is nothing different from the many' seem to me necessarily false given our ordinary understanding of identity and counting. Everything is identical to itself and to nothing else, in particular, nothing is identical to many things, each of which is different from it. If we measure commitment by the number of objects in our ontology, then a commitment to a cat-fusion is a further commitment, over and above the commitment to the cats which are its parts. If we have ten cats, then the cat-fusion which has all the cats as its parts is an eleventh object. How else could we measure commitment? Perhaps Lewis means that the cat-fusion is not a further commitment in the sense that, necessarily, if the cats exist, the cat-fusion exists. But this would be an unwise move. Since, for him, possible worlds are necessary existents, commitment to possible worlds would be no further commitment given a prior commitment to anything whatsoever. But surely even Lewis does not think his possible worlds are so innocent in the comforting sense he intends.

Sometimes I have the impression that Lewis means by the ontological innocence of fusions a thesis which is expressed in the following passage:



222 ALEX OLIVER

Describe the character of the parts, describe their interrelations, and you have ipso facto described their fusion... Its character is exhausted by the character and relations of its parts. ([5], p. 80)

On this reading, the innocence is not well-described as 'ontological'. Rather, the innocence is really an innocence from what have been called emergent properties, properties of composite objects which are not deter- mined by the properties and relations of their parts. Lewis's claim is that

mereological fusions have no such properties. If we suppose Lewis right, then one might think that further understanding of how a class is

composed of its singletons is gained by this ease in describing fusions. But this is wrong since Lewis says he knows next to nothing about the character of fusions with singletons as their parts ([5], 52.1). This last admission should make us question Lewis's claim that he understands how classes are

composed of the singletons of their members. Suppose I said to you that I understand how a molecule is composed of its atoms but that I know noth-

ing about the character of a molecule or an atom. You should complain that my understanding is grossly incomplete. There is only a meagre sense to understanding a relation (here a compositional relation) if the character of its relata are unknown. If Lewis falls back on the fact that he understands the compositional relation holding between a class and the

singletons of its members, in the sense that he knows its formal properties, then I say I can know these formal properties without having to think of the subclass relation as a special case of the mereological part-whole relation.

Indeed, I question whether mereology is as innocent as Lewis makes out, on this reading of innocence. Lewis says that proper classes exist, for

example, the proper class of all non-self-membered sets. This proper class is the fusion of the singletons of each non-self-membered set, each such

singleton exists, and Lewis's principle of unrestricted composition entails that their fusion exists too. On pain of paradox, this fusion cannot itself have a singleton. Is this not a peculiar property of the fusion, which other fusions of singletons do not share? Lewis will say that this is just one of the

mysteries of the singleton function. But it takes two to tango. It is surely arbitrary to pin the blame on the singleton function; it is an equally mysterious property of the fusion just mentioned that it fails to have a singleton.

Lewis's First Thesis is certainly not self-evident and his considerations in its favour collapse under scrutiny. I wonder if it is false.3

Gonville and Caius College, Cambridge, CB2 1TA

3 I am very grateful to Michael Potter and Timothy Smiley for helpful comments.



ARE SUBCLASSES PARTS OF CLASSES? 223

References [1] D. M. Armstrong, Universals and Scientific Realism, volume 2 (Cambridge:

Cambridge University Press, 1978). [2] Bernard Bolzano, Wissenschaftslehre (1837), translated as The Theory of Science by

R. George (Oxford: Basil Blackwell, 1972). [3] David Bostock, Logic and Arithmetic, volume 2 (Oxford: Clarendon Press, 1979). [4] Henry Leonard and Nelson Goodman, 'The Calculus of Individuals and Its Uses',

Journal of Symbolic Logic 5 (1940) 45-55. [5] David Lewis, Parts of Classes (Oxford: Basil Blackwell, 1991). [6] Alex Oliver, 'The Metaphysics of Singletons', Mind 101 (1992) 129-40. [7] J. R. Shoenfield, Mathematical Logic (Reading, Mass.: Addison-Wesley, 1967). [8] Alfred Tarski, 'Foundations of the Geometry of Solids' (1929), reprinted in his

Logic, Semantics and Metamathematics (Oxford: Clarendon Press, 1956), 24-9. [9] Alfred Tarski, 'On the Foundations of Boolean Algebra' (1935), reprinted in his

Logic, Semantics and Metamathematics (Oxford: Clarendon Press, 1956), 320-41.

A Yabloesque Paradox in Set Theory LAURENCE GOLDSTEIN

Stephen Yablo [12] has produced a beautiful paradox which, although in the Liar family, is not self-referential. The importance of this discovery should not be underestimated.' A long line of attempted solutions to the Liar, from the mediaeval restringentes tradition, through Russell's Vicious Circle Principle, and up to the present day have attributed paradoxicality to self-reference and, accordingly, have sought to defuse the paradox by banning, for some reason or other or none, some or all types of self-reference. Yablo's ingenious variant is Liar-paradoxical but isn't self-referential, so a whole wodge of 'solutions' is immediately relegated to the scrap heap.

I shall begin by producing a paradox in set theory which is inspired by Yablo's example. Then I shall offer a solution to this paradox, and show that a counterpart to that solves Yablo's original paradox. The latter solution is one that I have previously advocated for handling the Liar. It is satisfying to vindicate that solution in this roundabout way. As Russell ([11], p. 485) said, '[a] logical theory may be tested by its capacity for deal- ing with puzzles ... since these serve much the same purpose as is served by

1 Perhaps, however, the discovery should not have been entirely unexpected in the light of Saul Kripke's result that the incompleteness of arithmetic can be proved without making use of any self-referential sentences (see Goldfarb [4]).

ANALYSIS 54.4, October 1994, pp. 223-227. @ Laurence Goldstein



Are Subclasses Parts of Classes?

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