Buried Abstracta, or how not to be a Linguistic Nominalist.

[During the last few weeks, NewApps has hosted several inter-connected discussions about the relationship between physics, metaphysics, and philosophy of physics (and their histories). [See here and here as well as Jody Azzouni's original intervention here; Abe Stone responded to Azzouni here; I responded here; Jeffrey Ketland intervened here. Jody Azzouni, in turn, responds to Ketland below. The forthcoming chapter by Arntzenius & Dorr {herafter A&D} here. -- ES]

It’s been long recognized by many philosophers that—if one wants to—there are lots of ways to bury quantificational structure elsewhere in a logical formalism. It’s a relatively mundane technical fact, in particular, that one can trade off quantification over entities by fattening up one’s “ideology,” as it used to be called—the predicates that one coins to talk about those entities.

Jeffrey Ketland’s comment, by raising a version of this issue, forces me to mention explicitly considerations that I’ve been tacitly presupposing. Ketland’s point is that under certain circumstances a theory T that quantifies over a mixed domain (let’s call them abstracta and concreta) can be replaced by a theory T* that quantifies only over the concreta, and that this can be done so that T* is definitionally equivalent to T.

How is the trick managed? Well, new predicates* are defined over the concreta, predicates* which have truth conditions equivalent to those of the predicates of the old theory that held (mixedly) over the abstracta—and this is done in the fashion that Ketland indicates. It’s important to realize, however, that the project won’t work unless one has a free hand in coining new predicates* that hold of concreta. It won’t work if the predicates* one is allowed to coin are restricted in some way. The project, therefore, is relevant to nominalism, it bears on the philosophical question of nominalism (indeed on ontological issues generally), only if one has a free hand in coining new predicates*.

Linguistic nominalists—this is terminology I’m making up on the spot—are nominalists who are obsessed with quantification. ”Say whatever you like about concreta,” they advise, “make up any predicates you like—that’s okay as long as you quantify only over concreta.” Linguistic nominalists regard the issue of nominalism as largely settled, because of the technical results Ketland mentions.

Consider that nominalist, however, who is concerned with what’s really in the world and with what what’s in the world is really doing. The second clause is the clincher: to that nominalist we can’t make up predicates arbitrarily. The truth conditions of the sentences of the new theory, such a nominalists will say, must be rooted in how the objects in the world is. If they’re not so rooted, then the new predicates* have actually buried the unacceptable abstracta into their notation—the role of the abstracta can only be recognized by asking: What do these predicates* mean, and what are their truth conditions? And by learning that the answer to that question turns on the very injection that was used to define those predicates* in the first place.

Consider that ontologist, more generally, who has a broader view of what predicates are acceptable than a nominalist does—such an ontologist might allow certain predicates into the language and not others, because only certain relations and properties are really possessed by objects and not others. Again, the point should be made that “really possessed” is ultimately a point about truth conditions: the objects having such and such relations and properties is what is supposed to make certain predicates hold of them or not (and therefore is what makes certain statements about those objects true and not others). Such an ontologist isn’t going to think that one is just allowed to make up any predicates* and relations* whatsoever just because doing so enables the needed injection into the ontological domain in question.

A number of cans of worms have been opened up by these two last paragraphs because, of course, “rooted” is metaphor, and so is “how the objects in the world is.” But it should be clear that the kinds of technical results that Ketland is citing don’t put an end to the debate between nominalists and their opponents (and perhaps Ketland indicates this in the opening sentence of his post by putting quote-marks around “nominalized”); they just begin it. For there is a (justifiably) monstrously large literature, that worries about how certain predicates are to supervene on other ones (other more acceptable ones), or how they are to be reduced (in some appropriate sense of “reduce”), and so on. Certainly it’s possible (and I think likely) that if one motivates a select set of concreta-predicates on the basis of the metaphysics of what concreta actually do, and what relations they actually have (according to nominalists, anyway) that the many coined predicates* needed to carry out the encoding of abstracta (and their predicates) into those concreta will not be admissible.

In earlier posts I complained about the positing of intrinsic structure. The above discussion shows in a really simple way how this can happen: new predicational structure is posited. On one view, of course, this structure has to be “in” the concreta in order to explain why these predicates* hold of concreta. But it’s hard to see why one can be justified in thinking it’s okay to posit this new predicational structure just because it enables certain “nominalistic” programs to be carried out.

There’s this saying about bumps in the rug ….

I should also add this: A & D do much more than the coding mentioned above, as I indicated in earlier posts and as Ketland notes as well. They posit additional “spaces” of various sorts that they allow themselves to quantify over on the grounds that doing so is as nominalistically innocuous as the positing of spacetime points themselves. But I’ve already complained about this. --Jody