## 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.

### Similar content being viewed by others

## Notes

I use ‘attribute’ for tropes (or particular attributes) as well as universals, and ‘nominalism’ for nominalism about attributes. This view differs from nominalism about universals in denying the existence of tropes as well as universals. See, e.g., Williams (1953) for an analysis of statements involving abstract nouns that invokes tropes. (Although the so-called trope theory is often considered a version of nominalism about universals, Williams presents a version of reductionist realism about universals (ibid., 9f)).

Actualist quantifiers are those that (with respect to a possible world) range over entities that actually exist (in that possible world).

I use ‘render’ in a broad sense to cover statements that express truthmakers of other statements as well as statements that give paraphrases or analyses.

In his view, any true statement must have a truthmaker and the truthmaker of a true statement is what makes the statement true, “that

*in virtue of*which it is true, or that which*makes*it true” (ibid., 29), or an entity whose existence “*necessitates*the fact” stated by the truth (ibid., 30).By contrast, no possible carmine particular completely resembles a possible vermillion particular. But it is not necessary to assume this to conclude that (2′a) is false.

See Yi (2014, 622–5) for more about Lewis’s scheme.

This is a reformulation of his possibilia rendering: “Every carmine particular resembles any vermillion particular more closely than some carmine particular resembles some French blue particular” (ibid., 225), where the two occurrences of

*some*are meant to take scopes wider than*every*and*any*. He also gives its modal version: “A carmine particular must resemble a vermillion particular more closely than a carmine particular can resemble a French blue particular” (ibid., 225). This is meant to have the same truth condition as the possibilia rendering and has the same problems as those for the possibilia rendering discussed below.The same holds for (2*)–(4*) below.

In proposing the minima scheme, Rodriguez-Pereyra explicitly restricts it to statements about determinates.

See Rodriguez-Pereyra (2002, 59–61) for his notion of

*sparse property*.He says “

*x*and*y*resemble each other to degree*n*if and only if they share*n*properties”, where by ‘properties’ he means*sparse*(or*determinate*) properties (ibid., 65). See his subsequent discussion of degrees of resemblance among carmine, vermillion, and French blue particulars (ibid., 66). See also note 15.I.e., ‘Some carmine particular resembles some

*triangular*particular less closely than any carmine particular resembles any vermillion particular.’In his discussion of degrees of resemblance, he reaches the same conclusion. He says that the degree of resemblance between a carmine particular and a vermillion particular with no properties other than colors and shapes that share a determinate shape (e.g., circularity) is 1, not 0 only because they share the shape (2002, 66).

In his view, determinables are disjunctive properties composed of their determinates (2002, 48f), and no disjunctive property is sparse (ibid., 51).

Rodriguez-Pereyra (2002, 49) gives mass properties as examples of determinate and determinable properties.

I use ‘be

*n*kg’ as a predicate for particulars interchangeable with ‘be*n*kg in mass’ and ‘have a mass of*n*kg’, not as a predicate for mass properties interchangeable with ‘be identical with the mass of*n*kg’.I.e., ‘Some particular of 1 kg resembles some particular of

*n*+ 1 kg less closely than any particular of 1 kg resembles any particular of*n*kg.’Degrees of resemblance cannot be infinite numbers in Rodriguez-Pereyra’s view (2002, 173), but even cardinal numbers that include infinite numbers cannot form an infinite descending chain.

I say that a determinable

*R covers*a determinate*Q*, if*Q*is a determinate of*R*.Thanks are due to an anonymous referee for

*Erkenntis*for suggesting this response on behalf of Rodriguez-Pereyra. But I think the response conflicts with his notion of resemblance. As noted above, he says the notion is one that “accounts for sharing of*sparse*properties” (2015, 225; original italics), but the proposal relates resemblance to some non-sparse properties.This was suggested by James Davies.

Some might propose to characterize degrees of resemblance as

*ratios*between the numbers of properties of a suitable kind, as an anonymous referee for*Erkenntnis*suggests. While this proposal assumes that the relevant properties shared by particulars (and those that they individually have) are finite in number, one cannot maintain this assumption in specifying what properties to use to characterize degrees of resemblance to deal with instances of (M_{ n }). In any case, the proposal does not help to make the minima-scheme renderings of its instances true. See the “Appendix” (see also note 31).See also note 15.

The view contradicts (1′a)–(1′b) as well.

That is, if (

*x*,*y*) and (*z*,*w*) are two pairs of particulars, then either (a)*x*and*y*resemble each other more closely than*z*and*w*do, or (b)*z*and*w*resemble each other more closely than*x*and*y*do, or (c)*x*and*y*resemble each other just as much as*z*and*w*do.It is hard to see why the carmine–vermillion difference between A and C would amount to the 1-kg difference rather than, e.g., the 1-pound difference.

Some might object to this assumption, but we can run the same argument with examples for which the counterpart of the assumption is irresistible, such as statements about mass and volume (e.g., ‘Being 1 cubic meter resembles being 2 cubic meters more than it resembles being 1 kg’).

We can weaken this assumption: π

_{2}is as close to π_{3}as to π_{1}or closer to π_{3}than to π_{1}. Moreover, it is not necessary to assume that there are no other determinate Π values; the argument given below goes through even assuming that Π has additional determinate values between π_{1}and π_{2}or between π_{2}and π_{3}.Note that this does not depend on any assumption about degrees of resemblance (nor does it assume that the degrees of overall resemblance among pairs of particulars form a linear order). So (4) raises a problem for any account of comparative similarity that uses the minima scheme.

The proposal is meant to avoid the problem of infinite descending chain (see the last paragraphs of Sect. 2 and note 24). Ratios between natural numbers (unlike natural or cardinal numbers) can form an infinite descending chain (e.g., 1, ½, ¼, etc.), and one might argue that there is an infinite descending chain of degrees of resemblance among particulars because different particulars have different numbers of S-properties.

See notes 35 and 36.

By doing so, they might aim to deal with all truths of the form ‘Being 1 kg resembles being

*r*kg more than it resembles being*s*kg’ (where*r*and*s*are real numbers). It is not necessary to include all mass properties of the form. For example, one may include only those of the form*P*[*r*,*s*] where*r*and*s*are positive*rational*numbers (see note 35). To respect similarity among mass properties (or resemblance in mass among particulars), however, an S-property that covers two determinates must cover any determinate that lies between them. One cannot include properties with ‘gaps’ in determinates they cover, such as*being*1 kg*or*3 kg, which does not cover some determinates (e.g., being 2 kg) that being 1 kg resembles more than it resembles being 3 kg.Moreover, any such particulars share (and lack) as many MS-properties as those they individually have. If one includes all determinables of the form

*P*[*r*,*s*] matching real numbers, the relevant numbers of properties are uncountable numbers. One can avoid this by including only determinables of the form matching rational numbers. On this proposal, too, the relevant numbers are infinite.Suppose that A and B are particulars of 1 kg and 3 kg, respectively, that share no determinate properties. Then let B′ be a particular of 2 kg that has the same non-mass properties as B. Then A and B do

*not*resemble less closely than A and B′ do (B and B′ have the same number of S-properties, and A and B share (and lack) as many S-properties as A and B′ do).

## References

Armstrong, D. M. (1978).

*Nominalism and realism: Universals and scientific realism*(Vol. I). Cambridge: Cambridge University Press.Carnap, R. (1967).

*The logical structure of the world. Trans. R. George*. London: RKP.Jackson, F. (1977). Statements about universals.

*Mind,**86,*427–429.Lewis, D. (1983). New work for a theory of universals.

*Australasian Journal of Philosophy,**61,*343–377.Pap, A. (1959). Nominalism, empiricism and universals–I.

*Philosophical Quarterly,**9,*330–340.Price, H. H. (1953).

*Thinking and experience*. London: Hutchinson.Rodriguez-Pereyra, G. (2002).

*Resemblance nominalism: A solution to the problem of universals*. Oxford: Oxford University Press.Rodriguez-Pereyra, G. (2015). Resemblance nominalism and abstract nouns.

*Analysis,**75,*223–231.Williams, D. C. (1953). The elements of being, I.

*Review of Metaphysics,**7,*3–18.Yi, B.-U. (2014). Abstract nouns and resemblance nominalism.

*Analysis,**74,*622–629.

## Acknowledgements

The work for this paper was supported in part by a SSHRC research grant (Grant No. 435-2014-0592), which is hereby gratefully acknowledged. I presented the paper in a Canadian Philosophical Association meeting. I wish thank the audience for useful discussions. I also wish to thank Kevin Kuhl, Marion Durand, James Davies, Chung-Hyoung Lee, and two anonymous referees for *Erkenntnis* for useful comments on previous versions of this paper.

## Author information

### Authors and Affiliations

### Corresponding author

## Appendix

### 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.^{Footnote 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.)^{Footnote 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., (M_{2}) and its minima-scheme rendering, (M_{2}*):

- (M
_{2}): -
Being 1 kg resembles being 2 kg more than it resembles being 3 kg.

- (M
_{2}*): -
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 (M_{2}*) 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 (M_{2}*) 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*.^{Footnote 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).^{Footnote 35} If so, the degree of resemblance between two such particulars cannot be defined and (M_{2}*) fails to be true.^{Footnote 36}

## Rights and permissions

## About this article

### Cite this article

Yi, Bu. Nominalism and Comparative Similarity.
*Erkenn* **83**, 793–803 (2018). https://doi.org/10.1007/s10670-017-9914-2

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10670-017-9914-2