## Abstract

The paper aims to develop a resemblance theory of properties that technically improves on past versions. The theory is based on a comparative resemblance predicate. In combination with other resources, it solves the various technical problems besetting resemblance nominalism. The paper’s second main aim is to indicate that previously proposed resemblance theories that solve the technical problems, including the comparative theory, are nominalistically unacceptable and have controversial philosophical commitments.

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## Notes

Some of these problems were raised in Goodman (1966) against Carnap’s phenomenological version of resemblance nominalism; see also Carnap (1928, §§70 & 72). Writing in the mid twentieth-century, Price claimed that most metaphysicians of properties at the time were resemblance nominalists (1953, p. 13). Today the situation is reversed; Goodman’s discussion may well have been responsible for the decline in resemblance nominalism’s popularity.

For the sake of simplicity we focus on (unary) properties rather than relations.

The terminology is from Rodriguez-Pereyra (2002).

We won’t consider other objections to resemblance nominalism, for example the regress problem pressed in Russell (1912).

The mere intersections problem apparently does not arise, since the

*R*^{2}-theory seems incapable of defining the notion of being a resemblance class of degree*n*.Lewis is considering the best form that resemblance nominalism might take. He is not advocating the theory himself.

The set theory in question has all the individual particulars as its urelements (i.e. entities in the domain of quantification that are not sets). Ranks are as standardly defined in set theory: the set {

*a*,*b*} (where*a*and*b*are urelements) has rank 1, the set {{*a*,*b*}} has rank 2, and so on. A pair set is a set with two members. An*n*th-rank pair set is a pair set of rank*n*such that all its elements are pair sets, elements of its elements are pair sets, and so on for*n*steps until urelements are reached. The transitive closure of a set*X*is defined as the smallest set containing*X*and closed under the union operation. Thus a set’s urelements are the individuals (non-sets) in its transitive closure. All the set theory used in this paper may be found in Goldrei (1996).*X*is a superclass of*Y*iff*Y*is a subclass of*X*.Observe that if conjunctive properties never exist then Rodriguez-Pereyra’s definition is in fact logically equivalent (for finite cases) to the following simpler definition: a maximal resemblance class is a property class iff it is not the intersection of any proper maximal resemblance superclasses. This simpler definition also extends to cases of infinite resemblance, whereas the R*diff-definition does not, since a class’s net resemblance contribution is not in general equal to the difference between the sum of some infinite cardinals from some infinite cardinal (however one extends the notion of subtraction to the transfinite). We come back to this point in §5.

Of the resemblance nominalisms founded on predicates other than

*R*^{2}, I know of none technically superior to the one presented here. For example, in a section of a paper discussing Carnap’s*Aufbau*(1975, pp. 68–73), Eberle proposed founding resemblance nominalism on a three-place predicate, call it*R*^{E}, whose intended interpretation is ‘*x*_{1}exactly resembles*x*_{2}in a certain respect but not*x*_{3}’. In property terms: ‘there is some property*P*that*x*_{1}and*x*_{2}instantiate but*x*_{3}does not’. Eberle’s proposal does not solve the imperfect community problem. Consider the following four-particular communities. The first community consists of particulars*a*_{1},*a*_{2},*a*_{3},*a*_{4}, where*a*_{1}instantiates*F*_{2},*F*_{3}and*F*_{4},*a*_{2}instantiates*F*_{1},*F*_{3}and*F*_{4},*a*_{3}instantiates*F*_{1},*F*_{2}and*F*_{4}, and*a*_{4}instantiates*F*_{1},*F*_{2}and*F*_{3}. The second community consists of particulars*a*_{1},*a*_{2},*a*_{3},*a*_{4}, where*a*_{1}instantiates*G, F*_{2},*F*_{3}and*F*_{4},*a*_{2}instantiates*G, F*_{1},*F*_{3}and*F*_{4},*a*_{3}instantiates*G*,*F*_{1},*F*_{2}and*F*_{4}, and*a*_{4}instantiates*G*,*F*_{1},*F*_{2}and*F*_{3}. The second community is thus the first community with some extra*G*-instantiations tacked on. The first community is imperfect (the*a*_{ i }do not share a property) and the second is perfect (the*a*_{ i }share a property:*G*). But in both cases any three particulars stand in the relation*R*^{E}.We call this and other collections ‘classes’ to preserve neutrality on whether they are sets or proper classes. Since standard set theories with urelements take the class of urelements to be a set, it would not be controversial to assume that P, and indeed all the other classes mentioned in this paper, are sets.

One reason for believing in necessarily coextensive properties, that they appear in mathematics, does not sit well with the comparative theory. If mathematical properties such as

*being trilateral*and*being triangular*exist then any two classes presumably share properties such as*being a set*, contrary to an important motivating idea behind the comparative theory, that classes share the properties shared by their urelements. And if the property theory is intended to be a theory of sparse properties, that is another reason for thinking that mathematical examples don’t qualify, since mathematical properties don’t seem to be sparse. But this last claim is controversial. Sober (1982) argues that*being trilateral*and*being triangular*have different causal roles (a device could be designed to detect one but not the other) and are therefore distinct. There are other arguments for necessarily coextensive properties, for example that there could be a property determinate with just one determinable.Rodriguez-Pereyra (2002, pp. 175–176) contains an argument that presupposes his argument against a collective notion of resemblance, discussed in the next paragraph.

Given that he has to quantify over the subscript in ‘

*R*_{ k }’ in giving a definition of a maximal resemblance class of degree*n*as we saw at the end of §2, it is not even clear that this is a consistent position.This confirmation could be inductive, or by exhaustion of instances (e.g. the world might have infinitely many inhabitants, or it might have finitely many, some of whom are capable of supertasks).

As he puts it: “Resemblance Nominalism, as a theory about what makes particulars have the properties they have, is not based on any contingent feature of the world” (2002, p. 98). See also §6(b).

‘+’ here denotes cardinal addition.

λ-many of the

*a*_{ i }omit λ-many of the κ properties. Since λ < κ and κ is infinite, the smallest cardinal*x*such that*x*+ λ = κ is κ itself.In this case, all but one—the improper one—of the κ subclasses of the class made up of the

*a*_{ i }also collectively resemble.If some property classes have infinite degree of resemblance and no conjunctive properties exist then the alternative definition to the R*diff method—that a maximal resemblance class is a property class iff it is not the intersection of any proper maximal resemblance superclasses—will work.

If variably polyadic predicates are not acceptable, that would be another reason to reject Lewis’s theory based on the primitive

*R*^{L}.E.g. the four-term comparative resemblance relation Williamson (1988) sees as underpinning the notion of degrees of similarity has this property.

Of course for all that has been said, \( {\fancyscript{R}}^{+} \) (or R

^{+}) could be definable from a predicate that is nominalistically acceptable; but there is no reason to think so.Rejection of sets and rejection of universals (or more generally primitive properties) spring from different sources. For instance, a broadly empiricist acceptance of spatiotemporal (or immanent or

*in re*) universals goes naturally with a rejection of mathematical objects. Conversely, accepting abstract mathematical objects but not universals is a coherent philosophical stance, as exemplified by Quine and others. The difference between the two kinds of metaphysical stance is apt to be blurred by the modern tradition of giving the rejection of sets (and abstract objects more generally) the traditional name for the rejection of universals, viz. ‘nominalism’. But these two types of nominalism, as we are now accustomed to calling them, are different. Moreover, it should be clear that the philosophical work done by sparse properties is different from that of classes, a point made forcefully in Lewis (1983). Not only is resemblance nominalism logically compatible with acceptance of standard set theory, then, depending on the source of one’s nominalism it may also be naturally allied with it.It seems that the theory must also rely on set theory in another, less problematic way. When criticising Rodriguez-Pereyra’s resemblance nominalism we saw that resemblance nominalism cannot take the infinitely many degree predicates

*R*_{0},*R*_{1},*R*_{2},…, as primitive. Since it must be capable of expressing them in order to technically solve the companionship problem, it must define them. But any such definition apparently requires that a transfinite recursion be effected on an argument place of some predicate or other. If so, the ability to effect such a recursion is a necessary ingredient of any extensionally adequate resemblance nominalism.

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## Acknowledgments

Thanks to Gonzalo Rodriguez-Pereyra, journal referees, and members of the Sheffield Philosophy department, especially Dominic Gregory, for comments.

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Paseau, A. Resemblance theories of properties.
*Philos Stud* **157**, 361–382 (2012). https://doi.org/10.1007/s11098-010-9653-6

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DOI: https://doi.org/10.1007/s11098-010-9653-6