Counting on universals 
In A Quinean Critique of Ostrich Nominalism (March 2012) Bryan Pickel and Nicholas Mantegani argue that so-called ostrich nominalism is less parsimonious than realism about universals.

Here's the background: The realist asks what can explain the fact that (for example) all blue things have something in common. The explanation, the realist says, is that the blue things all participate or instantiate a further thing: blueness, a universal. The ostrich nominalist denies that there is any explanation required; at least some things are brutely similar.

It is natural to think that the realist gets more explanatory power (giving an ontological basis to similarity) at the cost of complexity (adding universals to the ontology). That is to say that the realist account is less parsimonious. Pickel and Mantegani argue the opposite.

They develop their argument using this schematic case: "Box World: There is a blue sphere. There is a green cube. There is an orange sphere. There is a blue cone."

They then count up how many basic sorts are in the nominalist and realist ontologies. For the nominalist, they count six:

N1: Exists x Blue (x)
N2: Exists x Orange (x)
N3: Exists x Green (x)
N4: Exists x Sphere (x)
N5: Exists x Cube (x)
N6: Exists x Cone (x)

The realist does not need separate elements for Blue, Orange, and all the rest. Rather, the realist just has the two-place instantiation relation. 'There is a blue thing' is regimented for the nominalist as Exists x Blue (x) but for the realist as Exists x IS(x,blue). Counting basic sorts for the realist, they count four:

R1: Exists x Universal(x)
R2: Exists x Particular(x)
R3: Exists x Exists y IS(x,y)
R4: Exists x Exists y IS(y,x)*

One might object to this accounting on the grounds that the realist ontology simply has more things than the nominalist ontology has got. The former has four particulars plus six universals, whereas the latter just has four individuals. They spend some time addressing this objection and distinguish between quantitative simplicity (having fewer things) and qualitative simplicity (having fewer sorts of things). The objection mistakenly attempts to apply Occam's razor along quantitative lines, where the principle ought to cut along qualitative lines.

I am sympathetic to their reply, so lets pass over that objection.

The greater objection is that they have not fessed up to all of the complexity in the realist account. The realist is not merely adding the four sorts enumerated in R1-R4. In order to regiment the original description of Box World, the realist must say things like Exists x (IS(x,blue) & IS(x,spherical)). Rather than merely adding some number of universals, this requires that the universals be named. Depending on our choice of logic, the name blue might be a constant or an implicit predicate. In either case, it is a further bit of basic machinery.

There are six named universals, so the realist requires six further basic elements. Contra Pickel and Mantegani, and pro the usual intuition, the realist has a less qualitatively simple account than the nominalist.

* Perhaps their list could be winnowed down. R3 and R4 might be merged together, on the grounds that instantiation is brutely relational. Then R1 and R2 might be defined in terms of instantiation; e.g., to be a Particular is just to be something that is in the first place of the instantiation relation. If my argument at the end is right, then this cutting and shaving won't be enough.

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