There is a second way that Moderate Resemblance Nominalists can meet the Hochberg-Armstrong objection. They could suppose that ordinary particulars are bundles of two sorts of things, namely, tropes plus bare particulars. Each bundle would contain many tropes but just one bare particular. The bare particulars have just one “job” to do: that of distinguishing one bundle from another exactly similar bundle. The bare particulars would be the ultimate source of numerical distinctness. Suppose, for example, that we have two tropes of scarlet red, S1 and S2, and suppose that these tropes belong to two different bundles, B1 and B2, each containing a different bare particular, P1 and P2. In this situation, the distinctness of S1 and S2 is grounded in the distinctness of P1 and P2. It is their connection to two bare particulars that makes S1 and S2 different. Thus, the truthmaker for the proposition that S1 and S2 are exactly similar is the pair {S1, S2}, while the truthmaker for the proposition that S1 and S2 are distinct is the quadruple {P1, S1, P2, S2}. We don't have an adequate ground of the distinctness of S1 and S2 without making reference to P1 and P2. We will consider this Bare Particular Theory (9.1T.1A.2A) in more detail in the next chapter.
There is a third response that Resemblance Nominalists could make to the Hochberg-Armstrong objection. They could posit distinctness tropes as the ground for the truths like (9), leaving the pair {A, B} as the truthmaker for the resemblance truth (8). In other words, what makes it true that A and B are distinct particulars is some distinctness trope D that connects them. There are two worries here. First, one might wonder how it is that a trope could connect two things if they are not already, prior to the connection, two things. The trope D is what is supposed to make A and B two, but how can it do that? If D is a binary trope by its very nature, it would seem that the twoness of its relata would have to be in existence in a way that is ontologically prior to D's actually connecting them. Second, if we need D to ground the distinctness of A and B, then that must be because neither A nor B nor the combination of the two can do this. That means that D has to be distinct from A, from B, and from the pair if it is going to be able to perform its ontological job. But this will require more distinctness tropes—one to distinguish D from A, another to distinguish it from B, and perhaps a third to distinguish it from the pair. In addition, each of these distinctness tropes will require still more distinctness tropes for them to do their job. We seem to be generating an infinite regress of tropes.
A fourth response to the Hochberg-Armstrong objection is to deny that propositions like (9) require a truthmaker at all. Some philosophers (including Ludwig Wittgenstein in his Tractatus Logico-Philosophicus) have argued that there are no identity or distinctness propositions at all or if there are that they are only about language, not about the world. This approach dissolves the need to ground (9), but it comes at a high cost. We would have to assume that it is absolutely necessary (even a logical truth) that the world contains exactly the fundamental entities that it does contain. We would have to rule out the possibility that any fundamental thing is contingent in its existence, or that there could have been more or other fundamental things than there in fact are. Whether we consider the fundamental things to be physical particles or fields, or more complex things like people or organisms, it seems obvious that many, if not all of them, exist contingently. If so, then truths of the form ‘There exist at least n distinct things’ should be treated as genuine assertions.
However, even if it is the case that there are truths about distinctness, we still have to ask whether these truths are among the truths that require truthmakers. If we were to accept Truthmaker Maximalism, we would have to find truthmakers for all truths, including truths of distinctness. However, we must consider restricted versions of Truthmaker Theory. Two plausible restrictions seem most relevant, namely, the restriction to positive truths, and the restriction to contingent truths.
Are truths about distinctness positive or negative truths? Consider (10) and (11):
(10) Mt. Everest is identical to Mt. Everest.
(11) Mt. Everest is distinct from K4.
Which of these is positive and which negative? It seems that ‘identity’ and ‘distinctness’ refer to contradictory properties, in the sense that the falsity of (10) corresponds to (12), and the falsity of (11) to (13):
(12) Mt. Everest is distinct from Mt. Everest.
(13) Mt. Everest is identical to K4.
If so, we could either take the logical form of (10) to be the denial of (12) or the logical form of (11) to be the denial of (13). That is, we could take (10) to be asserting the fact expressed in (14) or we could take (11) to be asserting the fact expressed in (15):
(14) Mt. Everest is not distinct from Mt. Everest.
(15) Mt. Everest is not identical to K4.
Suppose we took the identity statements to be positive. What would be the truthmaker of a truth like (10)? There would seem to be two possibilities. First, the truthmaker could be just Mt. Everest itself, or second, it could be some nexus between Mt. Everest and the property of self-identity. Either way, if identity statements were positive, then distinctness truths (like (11)) would be negative. (15) would be asserting the non-existence of a truthmaker for (13). However, both Mt. Everest and K4 do exist and both have the property of self-identity, so what possible truthmaker for (13) is in fact missing?
We seem to be forced to consider distinctness truths to be positive and identity statements to be negative. There is some truthmaker for (13), something that grounds the distinctness of Mt. Everest and K4. (12) is false because there is no ground for the distinctness of Mt. Everest from itself.
What about contingency-based restrictions of Truthmaker Theory? If statements of distinctness are necessary truths, they may not need truthmakers, even if they are positive. Are all distinctness truths necessary truths? Consider again (11):
(11) Mt. Everest is distinct from K4.
It seems unlikely that Mt. Everest could be identical to K4. If they were identical, they would be one and the same thing, and it seems plausible that any one thing must be one thing in every possible world in which it exists (see Kripke 1980 and Williamson 1996 for details). However, the fact that Mt. Everest and K4 couldn't be identical is not enough to make (11) a necessary truth, since we have to ask what would be the case if either Mt. Everest or K4 hadn't existed. It is at least arguable that in the absence of Mt. Everest or K4, (11) wouldn't have been true. If so, and if there are fundamental particulars that exist contingently (without necessity), then truths of distinctness involving such contingent particulars will not be necessary truths.
In addition, it is far from clear that no necessary truth has a truthmaker. It may be that trivially true statements or statements true by something like a stipulation, like ‘All bachelors are unmarried’, have no truthmaker. However, many necessary truths are far from trivial. Among the class of substantial necessary truths, some seem more fundamental than others. Truthmaker Theory can explain this fact by identifying the fundamental truths with those asserting the existence of a single, necessarily existing truthmaker.
2. Explaining the Facts of Resemblance. The second argument against Resemblance Nominalism starts with the observation that this view seems committed to large number of distinct, fundamental resemblance relations. How so? As we have seen, Resemblance Nominalists must in some way distinguish between different degrees of resemblance. This can be done in either of two ways. First, we could suppose that there are a large number of fundamental resemblance relations, each corresponding to a certain degree of resemblance, forming together a linear ordering of degrees. This is Gonzalo Rodriguez-Pereyra's (2002) approach. He posits an infinite number of degrees, one for each natural number. In addition, we have to suppose that it is metaphysically impossible for a group of things to resemble each other to different degrees at once. The second option is to suppose that there is a single, comparative resemblance relation: the relation of the x's resembling each other more than they resemble the y's. This was David Lewis's (1999a:
14) proposal.
It seems that Lewis's approach, employing a single comparative resemblance relation, is preferable in light of Ockham's Razor (PMeth 1). However, Lewis's proposal also brings with it a large number of necessary connections between distinct fundamental facts. For example, suppose that the members of X resemble each other more than they do the z's, and suppose that Y is a proper subset of X. Clearly, it must be true that the members of Y also collectively resemble each other more than they do the z's, yet the two collective resemblance-facts are equally fundamental, according to Lewis's theory. Neither can be grounded in the other.
Lewis's theory entails a second set of necessary connections, this time involving subsets of the comparison classes. Again, suppose that the members of X resemble each other more than they do any of the members of Y, and suppose that Z is a subset of Y. Again, it must be the case that the members of X resemble each other more than they resemble any of the members of Z, but this will have to be a brute necessity connecting the two fundamental facts.
In contrast, Realists have a ready explanation for both of these facts. If members of the X resemble each other in a way that they do not resemble the z's, then the members of X must each instantiate some universal that is not instantiated by any of the z's. But of course it then follows that all of the members of any subset of X must also instantiate some universal that is not instantiated by the z's, and that the members of X instantiate a universal that is not instantiated by any member of a subset of the z's.
There is a similar class of brute necessities required by Rodriguez-Pereyra's account. First, as we have seen, it has to be impossible for two things to resemble each other to two different degrees. Second, resemblance to any degree must be symmetric; that is, if x resembles y to degree n, then y must resemble x to degree n. Third, if the members of X resemble each other to a certain degree, and the members of Y resemble each other to a certain degree, and there are some things in the intersection of X and Y, then the things in that intersection must resemble each other to a greater degree than do the things that are in X but not Y, or in Y but not X. None of these things can be explained in terms of the instantiation of universals, and so they must be brute necessities. But, as we have seen, Ockham's Razor directs us to minimize such necessities (PMeth 1.2).
Resemblance Nominalists, in addition, must posit a large number of metaphysically fundamental similarity facts. If there are n distinct particulars (whether ordinary particulars or tropes), then there must be at least 2n fundamental facts about the comparative degree of similarity among sets of particulars (using Lewis's definition). In contrast, if there are n particulars and m universals, Realists require only n·m fundamental facts, a much smaller number (if n is at all large). In fact, Realists who embrace Totality Fact Maximalism require only m fundamental facts, one for each universal. If there are only m universals, then Realism entails that there can be only m natural classes. However, Resemblance Nominalists have no way of putting an upper bound on the number of natural classes because any class could be a natural one, as long as its members resemble each other more than they do any non-member. Thus, Lewis's version of Resemblance Nominalism requires a large number of independent, fundamental resemblance-facts. This counts against Moderate Resemblance Nominalism, since it is a failure of quantitative economy (according to Ockham's Razor).
If one attempts to avoid some of these difficulties by replacing Lewis's definition of resemblance classes with a class of binary relations of resemblance to a degree (as in Rodriguez-Pereyra 2002), there are still a number of necessary connections left unexplained. For example, Rodriguez-Pereyra cannot explain why resemblance to a degree is symmetrical, that is, why, if x resembles y to a certain degree, y resembles x to that same degree. Realists can easily explain this, since to resemble to a certain degree can be analyzed in terms of the number of universals instantiated in common. In addition, Rodriguez-Pereyra cannot explain why, if the pair {a,b} resembles the pair {c, d} to a certain degree, the particular a must resemble the particular c to at least that same degree. In addition, since Rodriguez-Pereyra has an infinite number of degrees of resemblance, his theory will require an infinite number of fundamental resemblance-facts.
SIMILARITY BETWEEN PROPERTIES: A PROBLEM FOR REALISTS? When evaluating the relative simplicity of Resemblance Nominalism and UP-Realism, we have to take into account one more factor, namely, degrees of similarity between properties themselves. The property of being red resembles the property of being orange more than it resembles the property of being green. A cubical shape resembles the shape of a rectangular prism more than it does the shape of a cone. Some Moderate Resemblance Nominalists already have a three-place relation of comparative similarity (x resembles y more than it does z) in place. They could use this same relation to define comparative similarity among properties: property A resembles property B more than property C if and only if the members of the resemblance class of A-tropes resemble the members of the resemblance class of B-tropes more than they do the members of the resemblance class of C-tropes. (Note, however, that this solution is not available to Resemblance Nominalists employing Lewis's definition of classes in terms of collective resemblance, since the members of the class containing both A-tropes and B-tropes will not resemble each other collectively in a way that they do not likewise resemble the C-tropes.)
Realists, in contrast, would seem to be forced to add a new fundamental relation, comparative similarity among universals, to their theory: the universal of REDNESS resembles the universal of ORANGENESS more than it does the universal of GREENNESS. If Realists take this route, then they increase their stock of qualitative primitives from two to three (comparative resemblance is added to being a universal and being an instantiation pair). This is a significant cost.
David M. Armstrong proposes that Realists provide some sort of reduction of comparative similarity to other fundamental terms. Realists might use a very sparse theory of universals, arguing that no two universals resemble one another at all. When two particulars resemble each other to some degree, this is always to be understood in terms of the two particulars (or their parts) instantiating or not instantiating the same universals. This would force Realists to deny that there are universals corresponding to the different colors or hues (like red, orange or green) or to the various shapes (cubical, conical, spherical). To be red must be to instantiate several, incommensurable color-universals which, when co-instantiated, constitute the color redness. Such a view does not square well with a common-sense view of our experience of these properties. In addition, it remains to be seen whether such a reduction can really be carried out.
As a second option, Realists could explain comparative similarity among universals by appealing to higher-order universals. If universals can instantiate other universals, then the similarity of two ground-level universals could be explained in terms of their sharing some third, higher-order universal. Perhaps two determinate shades, like REDNESS and GREENNESS, share a determinable universal, like SHADE OF COLOR. This would explain why REDNESS is more similar to GREENNESS than either is to CUBICALITY or any other non-color. To explain why REDNESS is more similar to ORANGENESS than to GREENNESS, we would have to introduce further higher-order universals, like the universal REDDISH COLOR, a universal shared by both REDNESS and ORANGENESS but not GREENNESS.
Third, Realists could move to a more abundant theory of universals, including generic universals (such as COLOR or REDDISHNESS) as well as specific universals (like PURE REDNESS) in their ontology. Comparative similarity could be a matter of how many universals two particulars share. An orange and a red sphere would be more similar to each other than either is to a green sphere because they share three universals (SPHERICALITY, COLOR, and REDDISHNESS), while each has only two in common with the green sphere (SPHERICALITY and COLOR).
Fourth, Realists could use a system of part-whole relations between universals to capture these similarity relations (this is an idea we will explore more fully in Chapter 10). If
two universals are similar, we could suppose that one is a proper part of the other. Two universals are very similar when one contains nearly all of the parts of the other. This approach is very plausible when dealing with determinable quantities or qualities that vary in magnitude or intensity along a single dimension. So, the universal 2 GRAMS IN MASS could be supposed to have the universal 1 GRAM IN MASS as a part, with intermediate quantities containing the one and being contained by the other. In the case of qualities that vary continuously in more than one dimension, like color, we could suppose that each color property involves instantiating one universal for each dimension, such as hue, intensity, and brightness.
Let's summarize the problems for Resemblance Nominalism, along with the possible solutions to those problems. We've discussed five major problems: contingent predication, co-extensive properties, companionship, imperfect community, and the Hochberg-Armstrong objection. We've also considered four versions or repairs to Resemblance Nominalism: Concretism (with Worldbound Individuals), tropes (Moderate Resemblance Nominalism), degrees of resemblance, and a variably polyadic resemblance relation. Here is a table with the results:
Table 8.1 Repairing Resemblance Nominalism
Concretism with Worldbound Individuals Tropes Degrees Polyadic Relation
Contingent Predication Yes No – –
Co-extensive Properties Yes, but with exceptions Yes – –
Companionship No Only with generic tropes Yes Yes
Imperfect Community No Only with generic tropes No Yes
Hochberg-Armstrong No Only with bare particulars No No
These solutions can be combined—no one excludes any of the others. For example, David Lewis's view incorporates Concretism, degrees of resemblance, and a variably polyadic relation, but rejects tropes. Gonzalo Rodriguez-Pereyra's theory has the same combination of solutions, although he replaces the variably polyadic relation with a binary relation between sets constructed ultimately from pairs of particulars. These theories dissolve all of the problems except the Hochberg-Armstrong objection, but at a significant ontological cost of adding merely possible things to our ontology and adding a large number of basic facts and of brute necessities.
The Atlas of Reality Page 27
