Socrates' paleness and Plato's paleness would be examples of two distinct tropes, one belonging to each philosopher, even when they are pale in exactly the same way. Ordinary particulars are things like Socrates or the Moon: particular things that have properties, not properties themselves.
Trope Nominalists are not the only philosophers who believe in tropes. There are also Trope Realists. Trope Realists believe in two distinct types of properties: tropes (like Socrates' paleness) and universals (PALENESS itself). Extreme Realists, in contrast, believe in universals but not in tropes, just as Extreme Nominalists believe in ordinary particulars but not in tropes. In Section 8.2, we focus our attention on Trope Theory. We argue that there are two competing conceptions of what tropes are like, namely, modular tropes and modifying tropes, and so two fundamentally different sorts of Trope Theory. We consider some objections to both.
8.1.1 Predicate Nominalism
We sometimes talk about properties. Indeed, sometimes we quantify over properties. That is, we use phrases of quantification that seem to require properties within their range of values, as in (1) and (2):
(1) Napoleon had all of the qualities needed by a great general.
(2) A circle has more properties in common with an ellipse than it has with a triangle.
Nominalists owe some account of what such phrases as ‘all of the qualities’ or ‘more properties’ stand for in (1) and (2) if there are no universals like GENIUS or ROUNDNESS. One historically popular account is Predicate Nominalism:
8.1T.1 Predicate Nominalism. Reductive Nominalism is true, and whenever two particulars resemble each other, their resemblance is grounded in the fundamental fact that the two fall under some one predicate of some language.
Predicate Nominalists, then, take properties to be predicates in some language (like English). Exemplification is just predicate satisfaction. This seems to be a case of conceptual grounding (as per Section 3.4), indicating that, for the Predicate Nominalist, there is nothing in the world corresponding to resemblance among particulars.
Predicate Nominalists might offer us the following paraphrases of (1) and (2):
(1′) All of the predicates in English that are necessarily true of all great generals are also true of Napoleon.
(2′) There are more predicates in English that are true of both circles and ellipses than there are predicates true of both circles and triangles.
A close cousin of Predicate Nominalism is Concept Nominalism:
8.1T.2 Concept Nominalism. Reductive Nominalism is true, and whenever two particulars resemble each other, their resemblance is grounded in the fundamental fact that the two fall under some one concept.
For Concept Nominalism, properties are concepts in some mind, something like predicates in a language of thought. Exemplification is falling under a concept. This is also a case of merely conceptual grounding (Section 3.4). Concept Nominalists could paraphrase (1) and (2) as:
(1′′) All of the concepts that apply with necessity to all great generals apply to Napoleon.
(2′′) There are more concepts that apply to both circles and ellipses than there are concepts that apply to both circles and triangles.
Clearly, these two accounts are structurally similar, and we treat them together.
A common complaint against both Predicate and Concept Nominalism is that they confuse the intrinsic properties of things with their extrinsic relations to a language or a mind. Things wouldn't cease to be spherical simply because the predicate ‘is spherical’ or the concept of sphericality ceased to exist. Things would still have been spherical even if there had been no languages and no concepts at all.
In addition, both seem to get the order of explanation wrong. A predicate like ‘is a circle’ applies to various concrete shapes by virtue of the fact that they are all circular. It's not the case that the shapes are circular by virtue of the fact that the predicate ‘is a circle’ is true of them.
8.1.2 Class Nominalism
The next version of Reductive Nominalism is Class Nominalism:
8.1T.3 Class Nominalism. Reductive Nominalism is true, and whenever two particulars have a property in common, this fact is grounded in the fundamental fact that the two belong to some one set or class.
Thus, Class Nominalism identifies properties with classes or sets. The property of being spherical simply consists in the set of spherical things. This has the advantage of identifying properties with things that have an existence that is independent of language or thought. Further, exemplification is just set membership; there is nothing more to exemplifying a property than being a member of the class that is that property.
There are, however, two serious types of objection to Class Nominalism. The first type results from Class Nominalism's identification of properties with classes of things. We will call this Class Nominalism's ‘extensionality problem’. This extensionality problem consists of two sub-problems: the problems of contingent predication and of co-extensive properties.
THE CONTINGENT PREDICATION PROBLEM Let's suppose that Class Nominalism is true, that properties are just classes of their instances. Here is a fundamental fact about sets: two sets are identical if and only if they have the same members. The set of dogs is different from the set of cats because the two sets have different members. And there is only one set of dogs. Any set with all of the dogs in it just is the set of dogs! The problem is that because of this connection between sets and their members, a set cannot exist if all of its members do not exist. But it seems that, in typical cases, a given property could have been exemplified by things that do not actually exemplify it. For instance, there might have been one more green apple than there is in fact, and thus one more green thing than there is in fact. So Class Nominalism seems committed to all three of (3–5):
(3) The property of being green is identical to a certain set.
(4) The property of being green, since it is a set, has its members essentially; any addition or subtraction of members is impossible.
(5) The set of green things might have had different members than it has in fact.
Unfortunately for Class Nominalism, (3–5) are inconsistent: they can't all be true together. The conflict is between the fact that sets have their members essentially and the fact that properties aren't essentially exemplified by just the things that happen to actually exemplify them. Thus, it seems that properties can't be sets of things as Class Nominalism contends. This is a serious problem, and it generalizes easily to a great many properties.
THE CO-EXTENSIVE PROPERTY PROBLEM Another extensionality problem emerges when we consider two distinct but co-extensive properties. Imagine a small world in which all and only electrons have negative charge. In such a world, the class of electrons would be identical to the class of negatively charged things, and so the two properties would also be identical. But the two properties are obviously not identical! For if they were identical, then it would be impossible for something to be negatively charged that wasn't also an electron. However, it is possible for there to be negatively charged non-electrons. So Class Nominalism gets the wrong result here as well.
One might try to solve this problem by combining Class Nominalism with Possibilism (12.1A.1T), and identifying the properties with classes of things, both actual and possible. The property of being spherical would be the class of all actual and possible spherical things, rather than just the class of all actually spherical things. There is a clear problem with this suggestion. Some things, like Pluto, are both possibly spherical and possibly non-spherical. If sphericality just is the class of all possibly spherical things, then Pluto has sphericality in all possible worlds, since it belongs to the class of possibly spherical things in all worlds. This would make Pluto necessarily spherical, which again is obviously wrong.
This problem can be avoided if we are willing to embrace David Lewis's theory of possible worlds as concrete universes (14.1T.1T), along with his doctrine of Worldbound Individuals (16.1A). On this view, the class o
f spherical things will not include our Pluto (the Pluto in our possible world), but it will include the spherical ‘counterparts’ (to use Lewis's term) of Pluto in other worlds. This certainly works, but it comes at the steep price of embracing Lewis's account of possible worlds.
THE PROBLEM OF THE SUPER-ABUNDANCY OF PROPERTIES The second difficulty with Class Nominalism is a consequence of the fact that it is a super-abundant theory of properties. Every class of things corresponds to a different property, and all such properties are metaphysically on a par with each other. This super-abundancy of properties leads to a series of paradoxes involving induction and causation.
One of the first such paradoxes to be proposed was the grue paradox of Nelson Goodman, his so-called ‘new riddle of induction’ (1954). Goodman defines a new color property:
(G) For all x, x is grue if and only if either (a) x was first observed before 2000 and is green, or (b) x was first observed after 2000 and is blue.
Every emerald observed before the year 2000 was observed to be green. Thus, all of these emeralds were also grue. Given the uniformity of experience, we had (before the year 2000) equally good reasons for accepting each of the following two hypotheses:
(6) All emeralds are green.
(7) All emeralds are grue.
However, the two hypotheses made contradictory predictions about what we would see when we uncovered the emeralds first observed after January 1, 2000. Hypothesis (6) entailed that these emeralds are green, while hypothesis (7) entailed that they are all blue. We would all agree that it is (6), rather than (7), that is the most reasonable hypothesis to embrace, apparently because ‘____is green’ is a more natural or conceptually simpler predicate than ‘___is grue’.
There are similar problems involved in the assignment of meanings to words and of contents to mental acts. For example, how do we know that the word ‘green’ in English doesn't correspond to a weirdly gerrymandered set of the kind introduced by Goodman's definition of ‘grue’ in (G)? How do I know that my concept of green corresponds to one set rather than another, when we try to extend the class of instances from those that we have actually encountered in the past to those we will encounter in the future?
To solve this problem, we have to make some distinction between the two properties. The sparse theory of universals offers a simple answer: some predicates correspond to universals, and most do not. There is no universal corresponding to the predicate ‘____is grue’, while there might be one corresponding to ‘___is green’. Even if there is no universal corresponding to either, we might be able to use universals to discriminate between the two classes. The class of green things probably has a simpler definition in terms of universals than the definition of the class of grue things.
Can Class Nominalists mimic a sparse theory of universals? David Lewis (1983) suggested that they can by introducing a distinction between natural and unnatural classes. The class of green things is natural, and the class of grue things is unnatural. Natural properties, properties of the kind that can figure in induction and in the interpretation of words and concepts, consist in natural classes only. Lewis suggests that it is a fundamental, irreducible truth about certain classes that they are natural.
There are two difficulties with Lewis's suggestion. First, it seems that Lewis's account has the order of explanation backward. What makes a class natural is the fact that it corresponds to the extension of some real property. It's not the case that what makes something a property is that it corresponds somehow to a class of the right kind. In addition, there is good reason to doubt that sets or classes have any natural properties or relations, except for the relation of membership:
PMeta 3 Membership the Only Fundamental Set-Relation. If S is a set, then the only fundamental relation involving S is the membership relation between S's members and itself.
A better solution for Class Nominalists would be to define the naturalness of a class in terms of the relations that hold among its members. Roughly, a natural class is one whose members resemble each other more than they resemble non-members. Green things resemble each other in color, but grue things bear no such resemblance to one another. This is a move toward Resemblance Nominalism, a view we'll consider presently.
8..1 Resemblance Nominalism
The final version of Reductive Nominalism that we take up is Resemblance Nominalism.
8.1T.4 Resemblance Nominalism. Reductive Nominalism is true, and whenever two particulars have a property in common, this fact is grounded in fundamental facts of resemblance between them.
The basic idea here is that at the fundamental level, there are just particulars, particulars which stand in metaphysically fundamental relations of resemblance. This is again a case of conceptual grounding: properties and property-exemplification are not to be found in reality: all we find there is a pattern of resemblance relations among particulars. For Resemblance Nominalists, if properties exist at all, they are constructed out of these particulars and identified with classes of resembling particulars (see Section 18.1.1 for more on logical constructions). More precisely, for each property P (other than the property of resemblance itself), Resemblance Nominalists propose that what it is to have property P is for a thing to resemble certain particulars, namely, those particulars in a resemblance class for P, where a resemblance class for a property is, intuitively, the set of things that exemplify that property (more precision below). Thus, if property P and property Q have the same resemblance class C, then the two properties are identical, since to be P is simply to resemble exactly the members of C, and to be Q is to resemble exactly the same things.
We can bring out the difference between Resemblance Nominalists and Realists by comparing their accounts of similarity and property-exemplification. Suppose that two particulars A and B resemble each other by way of having property P. Realists will say that the resemblance of A and B is grounded in the fact that they both exemplify the universal for property P. Resemblance Nominalists, in contrast, will take the resemblance to be fundamental, and they will suppose that each particular's exemplifying P is grounded in its similarity to certain particulars, namely, the resemblance class for P, a class that includes A, B, and all other particulars that resemble A and B in this same way.
There are two broad varieties of Resemblance Nominalism. In order to distinguish them and to see Resemblance Nominalism in more detail, we must have recourse to a few definitions:
Def D8.1 Property. A property is anything (whether universal or particular) that either grounds the character of something else or is implicated in a general account of attribute agreement.
Def D8.2 Ordinary Particular. An ordinary particular is a particular that is not a property.
Def D8.3 Trope. A trope is any particular that is also a property.
The nature of tropes is controversial, but Trope Theorists agree on at least this much: for any two things, let's call them a and b, which share some property P, there are two (non-identical) tropes of P had by a and b respectively. No trope, in other words, is had by more than one thing. In this sense, tropes are particulars. But tropes also ground character and are, in that sense, properties. We will return to tropes below (in Section 8.2), so we don't say much more about them here. On the basis of these definitions, we can distinguish extreme and moderate forms of Resemblance Nominalism:
8.1T.4.1T Extreme Resemblance Nominalism. There are only ordinary particulars, and whenever two ordinary particulars resemble each other, their resemblance is metaphysically fundamental.
8.1T.4.1A Moderate Resemblance Nominalism (Trope Nominalism). There are only particulars, and whenever two ordinary particulars resemble each other, their resemblance is grounded in the fundamental fact that the two are characterized by tropes whose resemblance is metaphysically fundamental.
To see how these views differ, and to get a bit clearer about the nature of Resemblance Nominalism, consider two red things, a ripe apple and a cardinal bird. Extreme Resemblance Nominalists claim that the property of be
ing red is a resemblance class of apples, cardinals, and other ordinary red things. A resemblance class is a set of things that bear strong resemblance relations to each other. The first philosopher to make use of the idea of a resemblance class was Rudolf Carnap in his Logical Construction of the World (1928/1967: 113). Here is Carnap's definition (he called these classes ‘quality circles’):
Def D8.4.1 Resemblance Classes (Carnap). A class of objects X is a resemblance class if and only if (i) each member of X resembles every other member of X, and (ii) nothing outside of X (that is, no non-member of X) resembles every member of X.
For Extreme Resemblance Nominalists, ordinary particulars are just unstructured wholes that primitively satisfy certain predicates in natural languages like English. Properties—that is, resemblance classes of ordinary particulars—do not, according to Extreme Resemblance Nominalists, ground character. The resemblance class of red things does not ground the character of red things. Properties are, however, implicated in a general account of attribute agreement. The resemblance class of red things is implicated in the Extreme Resemblance Nominalists' claim that two things are alike in some respect if and only if they are both members of some single resemblance class.
Moderate Resemblance Nominalists, on the other hand, claim that a red apple has the character that it does because it is related somehow to some tropes, particulars that are also properties. For Moderate Resemblance Nominalists, the (shareable) property of being red is a resemblance class made up of tropes rather than ordinary particulars. The property of being red is just the class of redness tropes; the property of being apple-shaped is the class of apple-shape tropes. These properties do not ground character (that is the job of tropes, after all), but they are implicated in a general account of attribute agreement, just as with Extreme Resemblance Nominalism. The precise nature of Moderate Resemblance Nominalism will become clearer in Section 8.2, where we turn our attention to tropes. However, the worries we will raise here beset both varieties of Resemblance Nominalism.
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