Before me are two white pieces of paper. In some ordinary sense, there is a property that these two pieces of paper share. The dispute between the realist and the nominalist is a dispute as to what sort of metaphysical account we should give of this very ordinary fact. According to the realist, we may give an metaphysical explanation of the fact that both pieces of paper are white: there is some one thing that is, in some sense, "in" both pieces of paper, viz. the universal whiteness. A universal is something that may "recur", i.e. be in more than one distinct particular, and which can explain genuine similarities between the particulars they are in, such as the fact that both pieces of paper are white.
(For the purposes of this discussion, we may say that for a universal to be "in" a particular is just for it to be part of it. Furthermore, the realist need not hold that there is a universal corresponding to every predicate that is true of more than one particular. E.g. we might explain the whiteness of each piece of paper in virtue of the fact that both have such and such microphysical structure; but in the end, according to the realist, we shall have to make recourse to universals to explain this fact.)
It should be noted that the problem is not to explain why the predicate "is white" is true of each piece of paper. That has an easy answer: "is white" is true of both pieces because they are both white. What the realist thinks universals are needed to explain is why both pieces are white, and why they are both rectangular, and why they both have such and such mass. This is Plato's ancient problem of One over Many: it is an argument from inference to the best explanation. According to the realist, the best explanation of the fact that both pieces of paper are white (or whatever) is that there is a single universal that is in each piece of paper.
The nominalist is one who denies that there are such things as universals. Two strategies for confronting the One over Many argument are open to the nominalist. One is simply to deny that there is any need for an explanation of why both pieces of paper are white (or that they have both have such and such microphysical structure). This is what Armstrong calls "ostrich nominalism" (having one's head in the sand) or "cloak and dagger nominalism." The other strategy is to offer some other explanation of why the two pieces of paper are both white, some explanation that does not make any appeal to universals. I will here look at two different sorts of alternative explanations, which Armstrong labels "class nominalism" and "resemblance nominalism".
Class nominalism is a doctrine that attempts to answer the One Over Many argument by appeal to classes instead of universals. Why is it that both pieces of paper are white? Because both are members of a certain class, viz. the class of all and only the white things. Classes are not "in" (part of) their instances, in the way universals are supposed to be; and furthermore, no matter what individuals you pick, no matter how dissimilar they are, there is a class containing all and only those things: this will not be the case with universals and their instances.
The first thing Armstrong notes about class nominalism is that its ontological commitment to classes might well be regarded as just as suspect as the realist's commitment to universals, so in and of themselves, there is no reason to prefer classes over universals. But of course, one might have an ontological commitment to classes anyway, as part of one's philosophy of mathematics, or part of one's semantic theory: and if you've got them in one part of your philosophy, you might as well use them to answer the One Over Many argument.
Armstrong's second point is that classes cannot account for the possibility of coextensive properties. The classic philosopher's example is, of course, being a creature with a heart vs. being a creature with a kidney. The class of creatures with a heart is identical with the class of creatures with a kidney: but, intuitively, these are different properties, and a thing has different causal powers in virtue of being a creature with a heart than it does in virtue of being a creature with a kidney. Of course, neither of these classes is going to correspond to a single universal, in Armstrong's view, but there are different universals involved in being a creature with a heart than there are in being a creature with a kidney.
This problem could be solved by using an ontology of possibilia in addition to an ontology of classes. Then the class of all possible creatures with a heart will not be identical with the class of all creatures with a kidney: for in other possible worlds there are creatures that have a heart but not a kidney, and creatures with a kidney but without a heart. But as Armstrong notes, an ontology of possibilia may well be regarded by the nominalist as "a remedy that is worse than the disease".
A further point, which follows from the observation that no matter what individuals you pick, no matter how dissimilar they are, there is a class containing all and only those things, is this: not all classes are relevant to determining that two things are, intuitively, of the same kind, or have the same or similar causal powers. What is needed is some way of specifying some classes as natural classes. This might be taken as primitive, or analyzed in some other way, perhaps by appeal to universals or appeal to resemblance. Of course, appeal to universals is out for the nominalist, and if appeal to resemblance is made, one might as well go ahead and be a "resemblance nominalist" (discussed below).
All these points are less than decisive. But Armstrong thinks he has an argument that is decisive: "the relation regress". If I explain the fact that both pieces of paper are white by appeal to the fact that both are members of a certain class C, Armstrong asks for an explanation of the fact that both are members of that class. If I try to explain this by appeal to further classes, by saying that <piece 1, C>, and <piece 2, C> are both members of some further class, Armstrong will again ask for an explanation, ad infinitum.
This is an argument to be resisted. As Wittgenstein said, in one of his few intelligible epigrams, "explanation has to come to an end somewhere", and a philosopher is not to be faulted for making appeal to some primitive notion. Armstrong himself cannot escape his own relation regress: whereas the class nominalist makes appeal to class membership, and leaves that unexplained, Armstrong makes appeal to the presence of universals, and leaves that unexplained. As Lewis writes in "New Work for a Theory of Universals," "Let all who have felt the sting of Armstrong's arguments rise up and say 'tu quoque'. "
Resemblance nominalism, rather than making appeal to class membership to explain sameness of kind, makes appeal to resemblance between things. Both pieces of paper are white in virtue of their resemblance to some paradigm white objects. (The paradigms might differ for different persons who possess the concept of white, but as long as the classes turn out coextensive, or close to it, this doesn't really matter.) Armstrong considers specifically a theory put forth by H. H. Price: both pieces of paper are white in virtue of the fact that both resemble the paradigms at least as much as they resemble one another.
One problem that Armstrong notes is the following: it might turn out that there are no suitable objects to serve as paradigms. Surely it is possible for only one object in to be of a certain kind (depending on the kind): but in such a case, there would be no available paradigms, and there would be no available explanation, in terms of resemblance to paradigms, of the object's being of that kind. Like the problem of coextensive properties for class nominalism, this could be solved by appeal to an ontology of possibilia. With possibilia, there are plenty of objects to serve as paradigms, no matter what was the case. But again, many would balk an such an ontology of possibilia.
But the argument that Armstrong considers his decisive one, is his "relation regress". If the nominalist tries to explain the whiteness of the pieces of paper by appeal to their resemblance to the paradigms, Armstrong asks in virtue of what do the pieces resemblance the paradigms. Again, this is an argument to be resisted. Armstrong does have a way of explaining resemblance, viz. by appeal to shared universals, that the nominalist cannot make; but although this does count in the plus column of Armstrong's theory, this is not decisive. The resemblance nominalist takes resemblance as a brute matter of fact, but everyone must take something as a brute matter of fact, even Armstrong.
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