[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.
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