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.
Wed 13 Jun 2012 02:16 PM
from: Bryan (Pickel)
Thanks for reading the paper! I was a bit worried about similar issues while working on it (and there are some remnants of these worries in footnote 13). I can’t speak for my co-author, but here’s my take.
You’re right that we were a bit coy about this in the paper about the names of universals. We do introduce names or haecceitistic (Pegasizing) predicates for properties. It may look like have to introduce them. But we don’t.
In particular, I see no reason to accept, in the general case, the assumption that:
"Rather than merely adding some number of universals, [regimenting the original description of Box World] requires that the universals be named"
I think that the Quinean view is that one can and should single out objects (including any universals) using the descriptive resources already available to one's theory. One should introduce a haecceitistic description only when the ability to characterize an individual is “so obscure or so basic a one that no pat translation into a descriptive phrase had offered itself along familiar lines.”
So, to turn the objection on it’s head: why should we introduce names for all of these universals unless we have to? That is, why should we accept the additional ontological commitments of assuming that every object posited by our theory is of its own primitive kind?
Relatedly, if we started introducing new primitive kinds for every individual we posit, that would eliminate many of the benefits of counting by kinds in the first place. (We would be no better off, for example, in comparing the commitments of theories that posit infinitely many things.)
I admit to not having taken sufficient care to work out the box world example. (Our primary concern was elsewhere.) But -- if the example were more realistic -- it would take argument to show that we need to introduce a name or haecceitistic predicate for each universal we posit.
There are, of course, standard objections to this sort of view. Our ability to single-out individuals using the existence descriptive materials might be limited, if there are, for example, symmetries. We might need some obscure "pegasizing" descriptions to break the symmetries. But it's not obvious that we need these descriptions for every object. There are also modal objections issuing from purported symmetries in the possibility space. But I'd need to see the details worked out.
Your objection might have something to do with the details of regimenting the ordinary English sentences. But I’d need to see a more specific argument that in order to regiment these sentences – to re-formulate them in a way that avoids problems but preserves what’s worthwhile – we need to introduce a name for every universal.
In sum, I learned to stop worrying and love the Quinean bomb.
Fri 15 Jun 2012 03:35 PM
Bryan, thanks for replying.
My thinking was just that the regimentation should be able to express the characterization of Box World. It's obvious how to do that using N1-N6, but R1-R4 seem too impoverished.
Your suggestion, if I understand correctly, is to be structuralist about the properties. So instead of Exists x (IS(x,blue) & IS(x,spherical)), it would be Exists x,y,z (Particular(x) & Universal(y) & Universal(z) & IS(x,Y) & IS(x,Z)).
This works, provided that a structuralist reconstrual of properties works. I don't have a strong opinion on the subject, but it's a point of philosophical contention. At the very least, it packages your defense of realism about universals together with structuralism about properties. And that's an extra commitment that's not explicit in the article.
Fri 15 Jun 2012 08:42 PM