The Atlas of Reality
Page 26
It should not be difficult to see that Extreme Resemblance Nominalism is faced with Class Nominalism's extensionality problems, both the problem of contingent predication and the co-extensive property problem. The reason they inherit these problems is that Resemblance Nominalism's theory of properties ensures that two properties are identical if they have the same resemblance class. Resemblance Nominalism entails the Resemblance Theory of Properties:
Resemblance Theory of Properties. Properties P and Q share the same resemblance class if and only if P and Q are identical.
For Resemblance Nominalists, what it is to have a property is simply to resemble certain things (the resemblance class for that property). If what it is to have P is the same as what it is to have Q, then P and Q must be identical. This means that no property can ever change its extension or have a different extension in different possible situations, since a difference in extension or resemblance class is ipso facto a difference in property identity. Similarly, it must be impossible for two distinct properties to share the same resemblance class.
The introduction of tropes enables Moderate Resemblance Nominalists to mitigate these two problems without embracing possibilia (merely possible things). For example, even if the class of red things and the class of round things were co-extensive, the class of redness tropes and the class of roundness tropes would be different (in fact, disjoint). Thus, the co-extensive property problem seems to be solved, at least in part. The only problematic cases remaining would be cases of co-extensive properties of tropes. For example, suppose that there are shape tropes that characterize plane surfaces: circularity tropes, triangularity tropes, and so on. Suppose that all of the triangularity tropes are also right-triangularity tropes (tropes grounding the character of being a right triangle) because it just so happened that all the triangular shapes were also right triangles. In such a possible scenario, the property of being triangular and the property of being right-triangular would be the same because the resemblance class of triangularity tropes would be identical to the resemblance class of right-triangularity tropes.
Tropes can also help with the problem of contingent predication, since tropes ground the characters they do essentially. A triangularity trope, for example, cannot become a squareness trope. When a concrete particular changes from being triangular to being square, this change is associated with the destruction of one trope and the creation of a new one. The tropes themselves do not change. However, this example brings out the fact that Moderate Resemblance Nominalism still has a problem with contingency and change, a problem we can call the problem of contingently existing instances. If, for example, at one time or in one possible scenario there are seven scarlet tropes, and at another time or in another possible scenario there are eight scarlet tropes, then Moderate Resemblance Nominalists are forced to recognize two distinct properties of being scarlet: one grounded in similarity to the seven tropes, and the other grounded in similarity to the eight tropes. This is clearly the wrong answer. There is only one property of being scarlet, a property that can vary in its set of instances from one time to another or one possible scenario to another.
TWO ADDITIONAL PROBLEMS: COMPANIONSHIP AND IMPERFECT COMMUNITY Nelson Goodman (1951) pointed out two additional problems for Resemblance Nominalism, problems that are not shared with Class Nominalism. These are the problems of companionship and imperfect community. Given Carnap's definition of resemblance class (Def 8.4.1), and the Resemblance Nominalist theory of properties, we get too few properties (the companionship problem) and too many properties (the problem of imperfect community).
THE COMPANIONSHIP PROBLEM Consider two properties P and Q that are distinct and have different resemblance classes, but A, the resemblance class of P, is a proper subset of B, the resemblance class of Q. That is, every instance of P is an instance of Q, but not vice versa. For example, P might be the property of being square and Q the property of being rectangular. In such a case, let's say that Q is a companion of P. If Resemblance Nominalism is true, then what it is to be P is to resemble all of the members of A. However, all of the members of B also resemble the members of A, since they all resemble each other and every member of A is also a member of B. To resort to our example, all of the rectangles do resemble all of the squares, since they resemble all rectangles and the squares are rectangles. Thus, it seems that every member of B should count as having the property P. But then the instances of P should include all of B, not just the members of the subset A. In other words, we would be forced to say that all of the rectangles are square! This would mean that the resemblance class of P is actually B, not A, making the properties P and Q identical after all.
The obvious solution of this problem is to tweak Carnap's definition of a resemblance class. Let's recall Carnap's original definition:
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.
The problem is that the class of square objects cannot satisfy this definition, since it fails condition (ii): some things outside the class of squares resemble all of the squares. In particular, the rectangles do. Why do we need condition (ii) at all? Why not simply drop it? If we dropped condition (ii) we would get too many resemblance classes and thus too many properties. Suppose that the shape of Times Square in New York is in fact square. Now consider the class of square things that are not identical to the shape of Time's Square. If we simply dropped condition (ii), then this class would also be a resemblance class, and being a square that isn't identical to Time's Square would be a property. Going down this route will result in a super-abundance of properties of the kind we wanted to avoid.
A better solution is to observe that the squares resemble each other more closely than any of the non-square rectangles resemble them. If we can replace the single, simple, binary resemblance relation with a family of resemblance relations (resembling to degree d, for some class of degrees) or a comparative resemblance relation (such as x's resembling y more than it does z), we can solve the companionship problem:
Def D8.4.2 Resemblance Classes (Comparative). A class of objects X is a resemblance class if and only if (i) each member of X resembles every other member of X to a certain degree, and (ii) nothing outside of X (that is, no non-member of X) resembles every member of X to that same degree.
THE PROBLEM OF IMPERFECT COMMUNITY However, there is an additional problem for Resemblance Nominalism, the problem of imperfect community. Suppose that there are three natural properties, being large, being green, and being cubical. Now consider the class of things that have at least two of these three properties. This class, call it At-Least-Two, will contain particulars of the following kinds:
Intuitively, there is no real or natural or “sparse” property that all of these things have in common. Nonetheless, the class At-Least-Two does satisfy the comparative definition of a resemblance class: any two members of At-Least-Two do resemble each other to a certain degree (by having two or three of the properties in common), and nothing outside the class resembles every member of the class to that same degree. At least, it seems plausible that we can find cases like this that do satisfy that condition.
Tropes help with the problem of imperfect community to some degree. In the case that we just considered, there would be three different resemblance classes, the class of largeness tropes, the class of greenness tropes, and the class of cubicality tropes. None of these classes form an imperfect community in the way that At-Least-Two does.
However, David Manley has pointed out (2002) that tropes alone do not entirely solve the problem of imperfect community, as long as all tropes are tropes of fully specific, determinate properties. Consider the class that includes all of the pink, baby-blue, and dark-purple tropes. These tropes all have two of three generic properties: the property of being pale (pink and baby-blue), the property of being bluish (baby-blue an
d dark purple), and the property of being reddish (pink and dark purple). Thus, they all resemble each other to a certain degree, and nothing outside the class resembles all of the tropes in the class to the same degree. Thus, this class also constitutes a comparative resemblance class, even though the tropes in the class do not (intuitively speaking) share any natural property in common.
The only solution available to Moderate Resemblance Nominalists involves positing a large number of new tropes, namely, generic tropes (tropes of mere paleness or mere bluishness or mere reddishness). If there are tropes of generic color then we could deny that the pink, baby-blue, and dark-purple tropes resemble each other at all: it is only thepale, bluish, and reddish tropes that resemble each other. Pink and baby-blue ordinary particulars resemble each other only by having paleness tropes, not by having pink or baby-blue tropes. However, the introduction of generic or determinable tropes comes with a high price tag, in two ways. First, this move requires a huge increase in the number of tropes. Each case of a colored concrete particular will contain a large number, perhaps even an infinite number, of color tropes, one for each kind of color property, however specific or generic it may be. Second, the move introduces a large number of necessary connections, for which there is no further explanation or ground. You cannot have a red trope by itself: it must always be accompanied by tropes of mere color, mere reddishness, and so on. However, a theory should ideally minimize the number of such brute necessities connecting separate entities (PMeth 1.2).
There is another route available to Resemblance Nominalists, whether Extreme or Moderate. This route involves a further modification of the definition of resemblance classes. What we want is for the members of the resemblance class to resemble each other in a certain way. The members of an imperfect community do not resemble each other in the same way. Dark-purple tropes resemble baby-blue tropes in an entirely different way from that in which the baby-blue and pink tropes resemble each other.
However, introducing new relations of resemblance in certain ways threatens to collapse Resemblance Nominalism into Ostrich Nominalism. Purple things and baby-blue things resemble each other by way of being bluish, while pink and baby-blue things resemble each other by way of being pale. It seems that to resemble each other in a certain way is just to have a certain property in common. If so, there are just as many ways of resembling things as there are natural properties. Hence, the Resemblance Nominalist will have no general explanation of why things resemble things, nor any general explanation of what it is to have a property in common. To have a property in common is just to resemble each other in the way that corresponds to having that very property. This is no explanation at all, and so this strategy collapses into Ostrich Nominalism.
There is, however, another way to go. We could abandon the idea that resemblance is a binary relation between two things. Instead, we could think in terms of the collective resemblance of a large number of things. The pale things are collectively in a relation of a certain degree of resemblance. However, the things that are pink, baby-blue or dark purple do not collectively resemble each other at all (or only to the minimal degree that all colors resemble each other). This is the route taken by David Lewis, who combines both collective and comparative resemblance:
Def D8.4.3 Resemblance Classes (Lewis). A class of objects X is a resemblance class if and only if (i) there are some z's such that the members of X collectively resemble each other and do not likewise resemble any of the z's, and (ii) X contains every y such that y and the members of X collectively resemble each other and do not likewise resemble any of the z's.
Instead of using a single comparative resemblance relation, we could also make do with a family of collective resemblance to a certain degree. Then we could say that a class X is a resemblance class just in case the members of X resemble each other collectively to a certain degree, and there is no y such that the members of X and y collectively resemble each other to the same degree.
This definition solves both the companionship problem and the problem of imperfect community, but at the cost of introducing a new, variably polyadic relation. By ‘variably polyadic’ we mean that the very same relation can hold between the members of a class X and the z's, no matter how many or how few things there are in either case.1
Besides these three extensionality problems, there are some further worries for Resemblance Nominalism. We turn now to these.
ADDITIONAL OBJECTIONS TO RESEMBLANCE NOMINALISM
1. The Hochberg-Armstrong objection. The first argument against Resemblance Nominalism comes from Herbert Hochberg (1999: 50–54) and David Armstrong (2004: 43–44). Consider these two propositions:
(8) A and B are similar fundamental particulars.
(9) A and B are distinct fundamental particulars.
What makes each of these statements true? For Resemblance Nominalists, the answer must be: the pair of A and B. In fact, not only do (8) and (9) have the same truthmaker in the actual world, this is the only truthmaker that either of them could have. In contrast, UP-Realists can say that (8) is made true by the fact that there is some universal U that both A and B instantiate and that statement (9) is made true by the pair {A, B}. In other words, (8) is made true by the two ordered instantiation pairs
(8) and (9) are clearly distinct in content. (8) could be true even if A and B were identical, and (9) could be true even if A and B were dissimilar. Neither proposition can be deduced a priori from the other. However, propositions that both have the same truthmaker and could have only that same truthmaker are metaphysically equivalent, that is, true and false in exactly the same situations, since it is part of the very essence of a proposition that it be true whenever one of its truthmakers exist and false if no truthmaker for it exists.
Resemblance Nominalists can reply by insisting on a distinction between a priori or conceptual equivalence and metaphysical equivalence. Although the conceptual contents of (8) and (9) do not guarantee their equivalence a priori, if as a matter of fact both (8) and (9) are true, then they must be true in exactly the same possible situations. This answer assumes two things. First, two entities that are numerically distinct in actuality are necessarily distinct, and second, two tropes that are similar are necessarily similar. Either of these claims could be challenged, but we will accept both of them for the sake of argument here.
The Hochberg-Armstrong objection suggests that there is something unsatisfying about this response. What's relevant about the difference between (8) and (9) is not just that they are not a priori equivalent. Instead, each attributes a different relational property to the pair of A and B. (8) attributes the property of distinctness, and (9) attributes the property of similarity. These properties are clearly different, as they have quite different extensions, even in the actual world. Some pairs of distinct things are exactly similar to each other, and some are not. Arguably, some “pairs” of exactly similar things are identical to each other, and some are not. Given the difference between the two properties involved, it seems natural to demand two distinct truthmakers, based on One Truthmaker per Fundamental Property:
PTruth 1 One Truthmaker per Fundamental Property. If p is the true predication of a fundamental property P to x1 through xn, and q is the true predication of a different fundamental property Q to the same things x1 through xn, then p and q have distinct truthmakers.
If we accept One Truthmaker per Fundamental Property (which we showed, in Chapter 3, to follow from a plausible definition of ‘fundamental property’), then we will need one truthmaker as the ground of the distinctness of A and B and another for their exact similarity. Resemblance Nominalists could meet this demand by supposing that the pair of A and B is the truthmaker for (9) but not for (8). This would require the postulation of a new entity as the truthmaker for (8). This additional entity would have to be a resemblance trope. If A and B are distinct and exactly similar, then the resemblance trope exists and has a particular i
ntrinsic feature corresponding to A and B's standing in the similarity relation.
However, once Resemblance Nominalists posit such resemblance tropes, they immediately fall into an infinite hierarchy of such tropes. If they adopt resemblance tropes, they must now give an account of the fact that two of these resemblance tropes are both of the resemblance kind. Given the constraints accepted so far, this will force Resemblance Nominalists to posit yet another, higher-order resemblance trope, standing between the two, resembling resemblance tropes. The same argument will force the positing of third-order and fourth-order tropes, and so on ad infinitum.
This infinite series comes at a very high cost in terms of quantitative economy. Moderate Resemblance Nominalists require an infinite number of tropes whenever there are two similar but distinct particulars. Does this also incur a high cost in terms of qualitative economy? Must Moderate Resemblance Nominalists posit an infinite number of kinds of tropes? It's hard to say definitively, but the various orders of tropes (first-, second-, third-, and so on) seem to involve different kinds of trope.
More importantly, the hierarchy of resemblance tropes constitutes an infinite regress, as Bertrand Russell (1912) argued. The existence of each resemblance trope R between A and B is supposed to be the ground of the truth that the two relata are similar, but this depends on the fact that the resemblance trope R is in fact of the resemblance kind, but this latter fact is supposed to be grounded in the further fact that the trope R is connected by resemblance tropes to the other members of the class of resemblance tropes. The truth of A and B's resemblance is never finally grounded. Its grounding is perpetually deferred.