Nominalism and Comparative Similarity

Abstract

Nominalism about attributes has serious difficulties in accounting for truths involving abstract nouns. Prominent among such truths are statements of comparative similarity among attributes (e.g., ‘Carmine resembles vermillion more than it resembles French blue’). This paper argues that one cannot account for the truth of such statements without invoking attributes.

Resemblance nominalists¹ hold nominalism about attributes, the view that no attributes exist.² This view has serious difficulties in accounting for truths involving abstract nouns. Prominent among such truths are statements of comparative similarity among attributes, such as (1)³:

• (1) Carmine resembles vermillion more than it resembles French blue.

In this paper, I argue that one cannot account for the truth of such statements without invoking attributes.

1 Comparing the Maxima

Lewis (1983, 348f) gives two related paraphrases of ‘Red resembles orange more than it resembles blue.’ They can be taken to give two versions of a scheme for rendering statements of the same structure as (1). Applying the two versions of the scheme to (1) yield (1′a) and (1′b):

• (1′)

  • a. Some carmine particular resembles some vermillion particular more closely than any carmine particular resembles any French blue particular.

  • b. A carmine particular can resemble a vermillion particular more closely than a carmine particular can resemble a French blue particular.

In (1′a), the quantifiers some and any range over possibilia. We can take this to make the statement equivalent to (1′b), a modal statement involving only the usual, actualist quantifiers.⁴ Rodriguez-Pereyra (2002), who holds nominalism, also proposes to render (1) as (1′a) or (1′b).⁵ Unlike Lewis, however, he does not take them to paraphrase or give analyses of (1). He does not think that it is necessary to give nominalistic paraphrases of true statements involving abstract nouns to defend nominalism; it suffices, he holds, to formulate nominalistic statements that “say what makes [such statements] true”, that is, nominalistic statements expressing their truthmakers (ibid., 92).⁶ Accordingly, he holds that (1′b) expresses “what makes (1) true”, and that this “is better expressed by” (1′a).

Lewis’s scheme has serious problems that Rodriguez-Pereyra’s modification inherits, as Yi (2014, 622–5) argues. Consider (2):

• (2) Carmine resembles vermillion more than it resembles triangularity.

The scheme renders this statement as (2′a) or (2′b):

• (2′)

  • a. Some carmine particular resembles some vermillion particular more closely than any carmine particular resembles any triangular particular.

  • b. A carmine particular can resemble a vermillion particular more closely than a carmine particular can resemble a triangular particular.

But these statements are false. (2′a) is false because a possible carmine particular completely resembles a possible triangular particular (the same particular might be both carmine and triangular)⁷; and it is the same with (2′b). So neither (2′a) nor (2′b) can be taken to paraphrase (2), which is true. Moreover, they cannot be taken to express the truthmaker of (2), either. A false statement has no truthmaker and cannot express the truthmaker of any true statement.⁸

2 Comparing the Minima

In response to Yi (ibid.), Rodriguez-Pereyra (2015) proposes another scheme for rendering comparative similarity statements. The possibilia version of this scheme renders (1) as follows:

• (1*) Some carmine particular resembles some French blue particular less closely than any carmine particular resembles any vermillion particular.⁹

(In this statement, as in (1′a), the quantifiers range over possibilia.)¹⁰ While Lewis’s scheme compares the maximum degrees of resemblance, the new scheme compares the minimum degrees of resemblance. (So call it the minima scheme while calling Lewis’s scheme the maxima scheme.) It is based on the idea that “if a determinate [property] resembles another determinate more closely than a third, the minimum degree to which something having the first determinate can resemble something having the second determinate must be greater than the minimum degree to which something having the first determinate can resemble something having the third determinate” (ibid., 226).

The new scheme, the minima scheme, renders statements of comparative similarity among (determinate) properties¹¹ as statements about comparison of degrees of resemblance among particulars. If so, what is the degree of resemblance between particulars? In formulating the scheme, Rodriguez-Pereyra uses “the notion of resemblance . . . that accounts for sharing of sparse properties” (ibid., 225; original italics).¹² On this notion, “resemblance holds between any two particulars sharing some sparse property” (2002, 64), and the degree of resemblance between them is the number of sparse properties they share (ibid., 65f).¹³

Now, in proposing the minima scheme Rodriguez-Pereyra assumes that (1*) and the scheme’s rendering of (2) [in short, (2*)]¹⁴ are true. (Otherwise they cannot express truthmakers of any statements.) But both renderings are false on the notion of resemblance he uses to formulate the scheme. He argues that (2*) is true because “the minimum degree to which a carmine particular can resemble a triangular particular (degree 0)” is smaller than “the minimum degree to which a carmine particular can resemble a vermillion particular (a degree greater than 0)” (2015, 225). But this argument rests on a false assumption: the minimum degree to which a carmine particular can resemble a vermillion particular is greater than 0. He supports this assumption by asserting that “no carmine particular can fail to resemble any vermillion particular” (ibid., 225). But a carmine particular cannot resemble a vermillion particular (on his notion of resemblance) unless they share a sparse property, and they might not share any such property. A carmine particular and a vermillion particular might share no non-color sparse property, and two such particulars share no sparse color property, either, because they have different determinate color properties.¹⁵ Although they share some determinable color properties (e.g., red), this does not help because (in his view) “determinables are not sparse properties” (ibid., 227, note 8).¹⁶ This means that both (1*) and (2*) are false.

Some might attempt to defend the minima scheme by taking sparse properties to include some deteriminables (e.g., red). This does not help. Consider statements about determinate mass properties of the following form¹⁷:

(M _n ):

Being 1 kg (in mass) resembles being n kilograms more than it resembles being n + 1 kilograms [where n is a positive natural number].¹⁸

All such statements are true. But not all of their minima-scheme renderings can be true, no matter what mass properties are considered sparse. Let d(n) be the minimum degree of resemblance between a particular of 1 kg and one of n kg. Then the rendering of (M _n )¹⁹ cannot be true unless d(n) is greater than d(n + 1) (It is equivalent to ‘d(n + 1) < d(n)’, for degrees of resemblance are cardinal numbers). So to make the renderings of all statements of the form true, the minimum degrees must form an infinite descending chain:

$$ {\text{d}}\left( 1 \right) > {\text{d}}\left( 2 \right) > {\text{d}}\left( 3 \right) > \cdots > {\text{d}}\left( n \right) > {\text{d}}\left( {n + 1} \right) > \cdots . $$

But this is impossible. The degrees of resemblance are natural or cardinal numbers (they are the numbers of sparse properties shared by particulars), and such numbers cannot form an infinite descending chain (for any collection of cardinal numbers has a smallest member).²⁰

Note that the above objection does not depend on any assumption about what kind of determinables are (or are not) included among sparse properties. Thus the objection equally applies to the minima scheme based on Rodriguez-Pereyra’s view that no determinable is sparse.

Instead of modifying his view of sparse properties, some might propose to characterize degrees of resemblance in terms of properties of a suitable kind that include sparse properties. (Call the properties used to analyze the degrees on this proposal S-properties.) They might then argue that the degree of resemblance between a carmine particular and a vermillion particular must be greater than 0 because some determinables that cover both carmine and vermillion (e.g., red)²¹ are S-properties although they are not sparse properties.²² But basing the minima scheme on the modified analysis of degrees of resemblance does not help to avoid the problem noted above. The scheme has the problem not because the degree of resemblance between particulars is given in terms of sparse properties, but because it is characterized as a cardinal number, the number of properties of a certain kind. While grounding all instances of (M _n ) requires an infinite descending chain of degrees of resemblance, numbers of properties cannot form such a chain. For cardinal numbers are well-founded, and any collection of them must have a smallest member. So the same problem arises for the scheme based on the modified analysis, for this analysis also characterizes degrees of resemblance as cardinal numbers (viz., the numbers of S-properties shared by particulars).

Some might attempt to avoid the problem by rendering (M_n) in terms of dissimilarity instead of resemblance:

(M_n**):

Some particular of 1 kg is more dissimilar to some particular of n + 1 kg than any particular of 1 kg is dissimilar to any particular of n kg.

They might then define the degree of dissimilarity between two particulars (e.g., one of 1 kg and one of 2 kg) as the number of S-properties that distinguish between them (e.g., being at least 2 kg).²³ But this proposal has the same problem in handling the mirror image of (M _n ):

(M_n′):

Being 1 kg resembles being (1 + (½)ⁿ⁺¹) kg more than it resembles being (1 + (½)ⁿ) kg.

To make all the dissimilarity renderings of instances of (M_n′) true, one would need an infinite descending chain of cardinal numbers as degrees of dissimilarity.²⁴

3 The Ordinary Notion of Degree of Resemblance

Some might argue that problems of the minima scheme noted above stem from inadequacies of Rodriguez-Pereyra’s notion of resemblance. They might then attempt to defend the scheme by basing it on an adequate notion or analysis of resemblance. I think his notion of resemblance is far removed from our ordinary notion and does not yield an adequate analysis of resemblance. But I do not think a correct analysis of resemblance helps to defend the scheme.

Rodriguez-Pereyra bases the minima scheme, as noted above, on a notion of resemblance on which the degree of resemblance between two particulars is the number of shared sparse properties so that they do not at all resemble each other unless they share a sparse property. Thus he holds that if three particulars with different determinate colors (e.g., carmine, vermillion, and French blue) have the same non-color properties, no two of them resemble each other more closely than they resemble the third (2002, 66).²⁵ This is not what we would say. If three particulars a, b, and c have the same non-color properties but are French blue, carmine, and vermilion, respectively, then “on our ordinary notion of resemblance, b and c resemble each other more closely than either of them resembles a”, as he puts it (ibid., 66). While saying that this is “what one is tempted to say” (ibid., 66), he rejects it to hold the opposite view stated above. As we have seen, however, his view contradicts even (1*),²⁶ which he assumes is true in proposing the scheme. If so, can one defend the scheme by reinstating the ordinary notion of resemblance?

I think that in the situation imagined above, the carmine and vermillion particulars resemble each other more than either does the French blue particular because the carmine and vermillion colors are closer to each other than either is to the French blue color. But this does not mean that there is an overall measure of resemblance on which comparability holds for any pairs of particulars,²⁷ for resemblance among particulars depends on features in multiple dimensions: color, shape, mass, etc. Suppose that A is a carmine particular of 1 kg, B a carmine particular of 2 kg, and C a vermillion particular of 1 kg (and that they have the same properties except color and mass properties). If so, A is closer to B in color than to C and closer to C in mass than to B. But this does not determine whether A has a higher degree of overall resemblance to B than to C or vice versa. In overall resemblance, A might not be closer to one of B and C than to the other. And this does not mean that A has the same degree of resemblance to B as to C.²⁸ If so, degrees of overall resemblance between pairs of particulars might form only a partial order: neither (A, B) nor (A, C) is a pair with a higher degree (of overall resemblance) than the other, although both have a lower degree than (A, A).

We can now see that basing the minima scheme on this ordinary notion of resemblance does not help to defend the scheme. Consider a true statement similar to (2):

• (3) Carmine resembles vermillion more than it resembles being 1 kg (in mass).

The scheme renders this as follows:

• (3*) Some carmine particular resembles some particular of 1 kg less closely than any carmine particular resembles any vermillion particular.

The scheme requires that (3*) be true. Assuming that it is, then, let a carmine particular (call it A) and a particular of 1 kg (call it B) witness its truth. We may assume that A has a determinate mass and B a determinate color.²⁹ So suppose, e.g., that A is 1 million kg and that B is black. Then A (being carmine) must resemble any vermillion particular whatsoever more closely than it resembles B. But this is false. A does not resemble a vermillion particular of 1 trillion kg (call it C) more than it does B. Neither (A, C) nor (A, B) is a pair with a higher degree of overall resemblance than the other, for neither the gap between the color differences (carmine vs. black and carmine vs. vermillion) nor the gap between the mass differences (1 trillion minus 1 kg and 1 million minus 1 kg) outweighs the other to yield a lower degree of overall resemblance.

Defenders of the scheme might argue that A does resemble C more than it does B because the gap between the color differences outweighs the gap between the mass differences. Some might perhaps hold that color and mass differences are commensurable in this way. But it is hard to see how those who do so can hold that the dissimilarity arising from the color difference between A and B (i.e., the carmine–black difference) cannot be outweighed by the dissimilarity arising from a far greater difference in mass between A and a possible vermillion particular (e.g., one of 1 quintillion kg).

Moreover, we can see that even postulating commensurability among degrees of resemblance in all possible dimensions of attributes does not help to defend the scheme. Suppose that Π is a physical quantity with three determinate values (π₁, π₂, π₃) that satisfies two conditions:

• (i) A possible particular has the quantity Π if and only if it has mass (0 or positive).

• (ii) The values of Π are linearly ordered with π₁ and π₃ at the opposite ends, and π₂ is the middle value that is as close to π₁ as to π₃.³⁰

Then consider (4) and (4*):

• (4) Being π₂ resembles being π₁ more than it resembles being 1 kg.

• (4*) Some π₂ particular resembles some particular of 1 kg less closely than any π₂ particular resembles any π₁ particular.

(4*) cannot express the truthmaker of (4), because it is false. To see this, suppose that a π₂ particular, A, and a particular of 1 kg, B, witness the truth of (4*). Then A must have a determinate mass property, and B a determinate Π value [by (i)]. So suppose, e.g., that A is 1 million kg and that B is π₁ or π₃. Then any π₂ particular must resemble any π₁ particular more than A does B. But this is false. Let B′ be a π₁ particular of 0.1 kg (that has the same sparse properties as B except in mass and Π). Then A and B′ are π₂ and π₁, respectively, but they resemble each other less closely than A and B do. (They resemble each other in Π as much as A and B do, but resemble each other less closely in mass than A and B do.)³¹

4 Concluding Remarks

We have examined two nominalist schemes proposed for comparative similarity statements, such as (1)–(4). The maxima scheme considers the highest degrees of resemblance between all possible particulars with one property and those with another, and the minima scheme the lowest degrees thereof. Both schemes fail, as we have seen. The maxima scheme fails, because particulars with two properties with little similarity (e.g., a color and a shape or mass property) might have the highest possible degree of resemblance (i.e., complete resemblance) because the same particular might have both of those properties. The minima scheme fails, because a property, P, might be more similar to another property, Q, than a third, R, while the lowest degree of resemblance between a P-particular and a Q-particular is not higher than that between a P-particular and an R-particular. It is straightforward to see this if we assume Rodriguez-Pereyra’s analysis of resemblance, on which the degree of resemblance between particulars is the number of properties of a certain kind (e.g., sparse properties) that they share. And we can see it without assuming the disputable analysis of resemblance. As the falsity of (4*) illustrates, any two particulars with less similar properties belonging to different dimensions (e.g., being π₂ and being 1 kg) might resemble each other more closely than some particulars with more similar properties belonging to the same dimension (e.g., being π₁ and being π₂), for the degree of resemblance between those particulars is not determined by the less similar properties but is affected by other properties they have (e.g., other mass properties and Π values).

And it is hard to see what else nominalists can propose to account for the truth of statements of comparative similarity among attributes. Some might consider rendering them as statements with ceteris paribus clauses, such as (1†):

• (1†) Other things being equal, carmine particulars resemble vermillion particulars more closely than they resemble French blue particulars.

But the clause does not help to handle statements comparing properties in different dimensions, such as (2). The ceteris paribus scheme renders (2) as follows:

• (2†) Other things being equal, carmine particulars resemble vermillion particulars more closely than they resemble triangular particulars.

But this is false. Just because a carmine particular, a vermillion particular, and a triangular particular have the same properties except color and shape properties does not mean that the carmine and vermillion particulars resemble each other more closely than the carmine and triangular particulars do. The latter might be identical or qualitatively indiscernible, but the former cannot. So I conclude that one cannot account for the semantics of comparative similarity statements without assuming the existence of attributes.

Appendix

To defend the minima scheme, some might propose to characterize degrees of resemblance as ratios between the numbers of properties of a suitable kind (in short, S-properties). On this proposal, the ratio proposal, the degree of resemblance between two particulars is the ratio between two numbers: (a) the number of S-properties they share, and (b) the sum of the numbers of S-properties they individually have.³² This proposal does not help to defend the scheme, either. For the ratio is ill-defined because both (a) and (b) would have to be infinite numbers. (Moreover, they would have to be the same infinite number.)³³

Call mass properties among S-properties MS-properties. Then proponents of the ratio proposal would have to include some (in fact, infinitely many) determinables among MS-properties. To see this, consider, e.g., (M₂) and its minima-scheme rendering, (M₂*):

(M₂):

Being 1 kg resembles being 2 kg more than it resembles being 3 kg.

(M₂*):

Some particular of 1 kg resembles some particular of 3 kg less closely than any particular of 1 kg resembles any particular of 2 kg.

If all S-properties are determinates, a particular of 1 kg might share no S-property whatsoever with a particular of 2 kg, which makes (M₂*) false on the proposal. To avoid this problem, some might include among S-properties the determinable property of being at least 1 kg and at most 2 kg (in short, P[1, 2]). They might then argue that (M₂*) is true because this is an S-property that covers both being 1 kg and being 2 kg, but not being 3 kg. So they might include among MS-properties all the mass properties of the form P[r, s] (i.e., being at least r kg and at most s kg), where r is a positive real number smaller than s.³⁴ In their view, then, any possible particular with a determinate mass (e.g., 1 kg) has infinitely many MS-properties (e.g., P[1, r] for any real number r > 1), and any two possible particulars with determinate mass (e.g., 1 kg and 2 kg) share infinitely many MS-properties (e.g., P[1, r] for any number r ≥ 2).³⁵ If so, the degree of resemblance between two such particulars cannot be defined and (M₂*) fails to be true.³⁶