METAPHYSICS UNIVERSALS - NOMINALISM LECTURE PROFESSOR JULIE YOO Varieties of Nominalism Predicate Nominalism The Nature of Classes Class Membership Determines Type Testing For Adequacy Primitivism Primitivist Analyses of Types Motivations for Primitivism Problems for Primitivism Coextension Problems ( Companionship Problem) Imperfect Community Problem Identity Conditions for Classes Order of Explanation Causal Argument Problem About Relations Higher-Order Types Resemblance Nominalism Advantages Over Primitivism Motivations for Resemblance Nominalism Constructing a Resemblance Classes Problems for Resemblance Nominalism Coextension Problem Imperfect Community Causal Argument Higher-Order Types and Relations Resemblance Regress: Resemblance Classes Require Universals Nominalism Lecture Page 1 of 9
NOMINALISM Varieties of Nominalism There is not doubt that there are types in the world properties and sortals that help to group the diversity of individuals into sensible groups. Consider this little universe: Individuals a = Abe s yellow lab b = Bob s yellow lab c = yellow pepper d = Eiffel Tower e = your left ear f = number 7 Types 1 = dog: {a, b} 2 = yellow: {a, b, c} 3 =??? {a, d, e, f} The question is how we are to explain the nature of types. According to Realism, types are universals. Nominalists, on the other hand, reject that there are universals, and explain the nature of types in one of two broadly different ways: 1) Predicate or Concept Nominalism, and 2)Class Nominalism. Predicate or Concept Nominalism are deeply flawed, because they cannot account for the fact that there can be types that go beyond the resources of human predicates and concepts, so we will not deal with them here, and only note that. It is Class Nominalism that will be the focus of our discussion on Nominalism. Class Nominalism explains the nature of types in terms of classes, and it comes in two main forms: Primitivism and Resemblance Nominalism. As the concept of classes is so central, we need to lay out some of their basic characteristics. The Nature of Classes Classes are just groupings of individuals. So {3} is as much of a class as {1} and {2}. The members of a class are called the class s extension. Thus, the extension of {1} is Abe s yellow lab and Bob s yellow lab. The identity of a class, then, is completely determined by its extension. Extension of a Class Determines Its Identity: Different members means different class; same members means same class. Thus, for example, if we added another yellow lab to the universe g = Cal s yellow lab and we formed a new class with it along with the members of {1}, we would have a new type: 4 = {a, b, g} This new class would differ from {1} even though both have members that are nothing but yellow labs. Now consider a another class {5}: 5 = {a, d, e, f} {5} has exactly the same members as{3}. In effect, we only have two labels {3} and {5}, but only one class. Nominalism Lecture Page 2 of 9
Class Membership Determines Types The Class Nominalist uses the apparatus of classes to define the nature of types. On this approach, individuals have the properties they have because of their membership in a class. That is, a and b are yellow labs they belong to{1}; a, b, and c, are yellow because they belong to {2}. Analysis of F-ness: F-ness just is some class C, where C is determined by its members. Analysis of x s having F: Some individual x has feature F because there is a class of individuals C such that x is a member of C. As you might have noticed, some classes are more natural than others. {4} is more natural than {5}. What differentiates the two flavors of Nominalism is how classes get constructed. Whereas the Primitivist says that the naturalness of a class is a primitive fact a fact that yields no further explanation the Resemblance Nominalist attempts to give an explanation of the naturalness in terms of resemblance. Testing for Adequacy One of the things that might strike you as odd in discussions of the problem of universals is the frequent appeal to very impoverished universes ones that only have a red cube, a yellow cube, and a green pyramid, or some such. The reason why these universes are constructed is because a theory of the nature of types needs to cover all possible cases. This is just standard practice for testing the adequacy of any philosophical analysis. Because it is imperative to examine whether there may be counterexamples to a given analysis, we need to consider such potentially problematic worlds. PRIMITIVISM According to Primitivism, a type just is a class of individuals, and having a property is nothing but being a member of a class. The fact that some classes are more natural have members that more intuitively form the class is a primitive fact. Primitivist Analysis of F-ness: F-ness just is some natural class C, where C s naturalness is a primitive fact. Primitivist Analysis of x s having F: Some individual x has feature F because there is a natural class C such that x is a member of C. Some classes display greater distributive unity than others (compare {4} and {5}). The notion of naturalness, however, is not precise. It cannot be ordered or ranked in a strict fashion. However, there certainly is content to the notion. Using Plato s metaphor, the aim is to carve the beast of reality along the joints. Scientific Realism takes this metaphor to heart. According to this view, science is the enterprise that attempts to carve reality at its joints, and introduce classes into our scientific and theoretical vocabulary that go beyond folk classes ( Fish are swimming creatures. Whales are fish. ) and reveal the true classes ( Fish are creatures with gills, Whales are mammals. ) Nominalism Lecture Page 3 of 9
For Quinton, a proponent of Primitivism, the notion of a natural class is in part an epistemological notion: as we encounter more samples of a given type, we learn to group them into a class. But this is, at best, an explanation of how we recognize certain classes as more natural than others; it should not be an account of what it is that makes a class natural. Since it is a primitive fact, we should not be offering explanations. Motivations for Primitivism A general motivation for going Nominalist is to pare down one s ontology. A theory that can explain the nature of types without appealing to entities one already acknowledge the individuals themselves and nothing but individuals is surely a p[referable theory. We already saw that Class Nominalism is better than Predicate Nominalism, since it can acknowledge types that go beyond our ways of classifying objects in the world, as classes are mind-independent. The greatest support for Primitivism, though, is negative, in the sense that it draws its strength from negative criticisms of Realism. Basically, the Primitivist complains that nothing is gained in the explanation of types when one examines the Realist account. To say that the paper in front of me is white because the universal WHITENESS is instantiated by the paper, does not accomplish more than merely saying that the paper is white because it belongs to the class of white things. The appeal to WHITENESS does not make progress. One might want to compare Primitivism with the Big Bang account of the origin of the universe that makes the appeal to God superfluous. PROBLEMS FOR PRIMITIVISM Coextension Problems ( Companionship Problem) Being a renate and being a cordate are two different types, but because renates and cordates happen to be coextensive belong to the same class the primitive natural class theory cannot accommodate their difference. Lewis attempts to solve this problem by appealing to possible class differences for contingently coextensive classes, so that we postulate a world where there are renates but not cordates (and vv), so that the possible difference of extension can be maintained after all. But this approach only goes a small distance, because some classes are necessarily coextensive, such as being round and being circular, or being triangular and being a three-sided enclosed figure. Imperfect Community Problem Here, the problem is that there can be a class but not a type corresponding to the class. This is the problem of class {3}. Remember, any grouping of individuals is a class. When we get to resemblance nominalism, we ll see this problem come up again. Consider the following class: a wooden square, a red square, and a piece of red wood. This class has members that resemble each other in some respect but not resemble others in the class in that same respect, so that there is no common type that unifies all the members. Nominalism Lecture Page 4 of 9
Identity Conditions for Classes This objection should reveal how Nominalism is committed strictly to nothing but particulars. A class is determined by its actual members, so that the class C of electrons e 1, e 2, and e 3, is a different class from the class C* of electrons that has e 2, e 3, and e 4. This is just a consequence of the way classes are defined by their members. Different members means different classes, even if the members exactly resemble each other perfectly. But surely we are dealing with only one and the same type. Order of Explanation It is more natural to say that something belongs to the class of electrons because it is the sort of thing that is an electron. Primitivism, however, reverses this order of explanation. It says that something is an electron because it belongs to the class of electrons. But compare with what you get with the Realist account: something is an electron because it instantiates the universal ELECTRONHOOD. Is this that much better? One might think that the Realist account sounds kind of empty. Causal Argument When a cause c brings about an effect e, it seems that the relation between c and e is a local affair involving only the intrinsic features of c and e. Say that you were nourished by eating a certain banana this afternoon. The feature of being nourishing is a feature that belongs to that particular banana. However, on a Class Nominalist analysis, that banana has the property of being nourishing because it belongs to a huge class of nourishing foods the bananas that were eaten last week, the pineapples in Brazil, the ham sandwiches that will be eaten in Paris next year, and so on. A class analysis of types ends up involving all the members of its class. And because something has the property it has because of its membership on a class, this means that all these other foods that having nothing to do with the banana you ate were involved in the banana s being nourishing this afternoon. But surely those other members have nothing to do with the particular causal relation in question. Problem About Relations Ordered classes cannot be the appropriate reduction base since they invoke order from the ground level. Primitivism, remember, is trying to reduce relations to classes. But mere classes cannot deliver non-symmetric relations, like being shorter than. Classes of classes won t do either, because mere classes do not impose an order upon its members. Ordered n-tuples cannot be the appropriate reduction base since they invoke relations from the ground level. The Wienerm-Kurotowski method, which tries to build asymmetry into classes by isolating the relevant members {<a,b> = {a}, {a, b}} only get us representations of relations, not the relations themselves. Higher-Order Types We have higher-order types, such as things like having mass or being colored, as well as relations between types, such as orange being more like red than like blue. Relations between types are crucial Nominalism Lecture Page 5 of 9
to the formulation of many laws of nature (example: G = m 1 m 2 /d 2 ) But Primitivism cannot easily handle these higher-order types. Consider the following a higher-order type: Redness is a color. The problem is that each red thing is not only a colored thing, it is also an extended thing. Thus, the class of all red things gives us not only the type RED, but also the type EXTENDED. There is also a problem for getting at relations between types: Redness is more like orange than like yellow: Depending on what individual things happen to be red, orange, and yellow, we might come up with a class that simply fails to deliver the appropriate analysis for the statement. Consider the case where the red thing and yellow things are apples. RESEMBLANCE NOMINALISM This is considered the most reasonable and promising version of nominalism. RM claims that x and y fall under the type F because they resemble each other with respect to F. Of course, the view cannot be stated this way on pain of circularity, because the appeal to the idea of being similar with respect to feature F is an idea that invokes a universal F-ness. Rather, the with respect to F needs to be based upon a the appropriate class. RN Analysis of F-ness: F-ness just is some natural class C, where C s naturalness is a analyzed in terms of the resemblance between the members of C. RN Analysis of x s having F: Some individual x has feature F because there is a resemblance class C such that x is a member of C. Just like naturalness, resemblance comes in degrees, and there is both a highest possible degree of resemblance, such as a class of electrons, and lower limit with very little resemblance, such as {5}. I say very little, rather than none, because all individuals can be said to resemble each other in some respect occurring after 500 AD. Motivations for Resemblance Aside from the ones that were already voiced by Primitivism ontological economy and being no worse off than realism RM has an advantage over Primitivism because it grounds the naturalness of a class in the objective resemblances between the particulars. It also avoids the problem of identity conditions that if a class loses or gains a member, then we have a different class because RM appeals to the concept of a resemblance structure rather than of a class simpliciter. On a resemblance structure, it does not matter how many members are in that structure; as long as the resemblances stay the same, the types stays the same. RM is also allege to have the following advantage over Realism. Price offers the following argument against Realism. Nominalism Lecture Page 6 of 9
1. If there are universals, then exact resemblance ought to be very common. 2. But exact resemblance is not very common. 3. There are no universals. He supports (2) by drawing our attention to the fact that different white things are of different shades of white. Different maple leaves differ in their size, color, shape, and structure. Realists reply by noting that there are kinds being a leaf, a tree, a human, and so on. We can assume that none of the instances of these kinds are exactly the same. However, there is some one thing, it seems that they all have in common. Therefore, exact resemblance is more common than Price claims. It seems that Price focuses only on determinates, so to speak, and not on the determinables under which the unique determinates fall; even if all shades of white differ from each other, ranging from the minute to the significant, they are all shades of white. Realists, on the other hand, focus on the determinables, and minimize the significance of the variations that may occur among determinates. Properties of Resemblance The relation of resemblance, however, is often a matter of degree, where we can go from little resemblance a tree and a sneeze to exact resemblance electrons. Resemblance is reflexive, symmetric, but not transitive. You resemble yourself (reflective), and if you resemble your sister, then your sister also resembles you (symmetric). But just because your sister resembles your mother does not mean that you resemble your mother (not transitive); this can happen when resemblances track different similarities. An exact resemblance class is an equivalence class, where all the members within the class C bear R to each other and to no other thing outside of C. For any other C*, there is an R* for the resemblance relation among the members of C*. An equivalence class is made up of members that satisfy the following relations: symmetry, reflexivity, and transitivity. Resemblance is an internal relation. That is, whether individuals x and y resemble each other is solely a function of their intrinsic natures that holds in all possible worlds. Constructing a Resemblance Class Paradigm Method: Pick an object(s) that is a paradigm representative of the feature you want to analyze, then construct the class around the object(s). Question: On what basis is the paradigm selected? Maximal Class Method: A maximal class is the class that has all and only those particulars with the feature you want to analyze. The class of blue things have all and only the blue particulars, and so on. Question: red particulars still resemble orange particulars; isosceles triangles still resemble scalene triangles, and so on, but they are still members of different types. Nominalism Lecture Page 7 of 9
Lowe s Account: The sort of resemblance class we want is one where the class of Fs is a maximal class of particulars such that, any two of them resemble each other at least as much as either of that pair resembles any particular which is not a member of the class. (see Lowe ch. 19) Price s Account: For any candidate d, d belongs to the class of a, b, and c, if and only if d resembles a, b, and c, as much as a, b, and c, resemble each other. Price s specifies two dimensions of resemblance: intensity: How similar the features are between particulars. extent: How similar particulars are as wholes Two individuals x and y that have a highly similar shade of white have a high intensity of resemblance, but if x is a piece of paper and y is a scoop of ice cream, then x and y do not resemble each other very much in extent. It would seem that the extent of resemblance between particulars depends upon the 1) the number of shared features between the particulars, and 2) the intensity of the resemblance of the features of the particulars. Lewis s Account: Some x resembles y more than x resembles z, or more specifically, x 1, x 2, resemble one another and do not likewise resemble any of y 1, y 2,, (where the strings of variables may be infinite and even uncountable) The individuals x, y, and z, refer to possibilia. PROBLEMS FOR RESEMBLANCE NOMINALISM Coextension Problem (Companionship Problem) In a class of co-extensive properties, there is one class of resembling members, and yet two types. The types triancularity and trilaterality are certainly different, but in a world with only triangles, this difference in type cannot be honored. Imperfect Community Here, there is a class of exactly resembling members, but no natural corresponding type. Suppose a world contains only: a red wood thing, a square wood thing, and a red square thing. Then these particulars are such that for each particular, it resembles every other member and nothing else. But surely there is no natural type corresponding to the class. Causal Argument One might think that RN can attribute the causal powers of a particular to the nature of the particular itself, but it actually cannot. All we get is the claim that objects act qua resembler of certain things (nourishment), rather than qua resembler of other things (being yellow). And when we make the qua specification, we necessarily bring in all the other members of the resemblance class that have nothing to do with that particular banana s being nourishing. Nominalism Lecture Page 8 of 9
Furthermore, in cases where properties are coextensive, such as being a renate and a cordate, or electricity and magnetism, both properties invariably get classed together, illegitimately involving both of them. Higher-Order Types and Relations Again, RN faces the same difficulties faced by Primitivism with regard to these issues. Concerning relations, mere resemblance does not give us order, which is what relations are all about. And concerning higher-order types, statements about types, as we saw, cannot be reduced to statements about individuals. Relation Regress: Resemblance Classes Require Universals Resemblance between objects need to be specified in terms of the respect in which they resemble marbles resemble each other with respect of shape, not color; snowflakes resemble each other with respect to color, not shape; and so on. But the introduction of the notion of there being some feature with respect to which objects resemble looks like the introduction of a universal. In the attempt to avoid universals, the RN appeals to relations of resemblance between particulars. However, when x and y are said to resemble, they can be said to resemble only in virtue of the existence of a single type of relation, namely, that of resemblance. But this single type of relation is a universal. R 1 R 2 a b c d Suppose a and b are cats, and c and d are instances of middle C. While only a and b resemble each other in being cats, while c and d resemble in being sounds at the same pitch, there is something that is similar between the two classes namely, that they have resembling members. Notice that even if we said that R 1 and R 2 are not themselves of one and the same type, there would still have to be a further type R 3, as Russell points out, in virtue of which R 1 and R 2 are instances of the resemblance relation. Reply: Appeal to exemplars. Rather than saying that x and y resemble in respect of F, say instead that there are exemplars of F towards which x and y resemble. Price, here, is offering an account of how to construct a resemblance class. They way to do it is by picking exemplars and then building the class around particulars that resemble the chosen exemplar: What is required is only that every other member of the class should resemble the class exemplars as closely as they [the exemplars] resemble one another. (p. 31) Suppose that the exemplars are a (a brick), b (ripe tomato), and c (fire truck). Each of these things resemble on another. For any candidate d, d belongs to the class of a, b, and c, if and only if d resembles a, b, and c, as much as a, b, and c, resemble each other. Question: But how are the exemplars chosen? Can they be chosen without presupposing the very thing RM is trying to avoid the appeal to universals? After all, a, b, and c can be exemplars of a feature F only if it is already identified ahead of time that F is the respect in which they resemble. Nominalism Lecture Page 9 of 9