In “How to Define Theoretical Terms” (1970), David Lewis says the following. Take a theory T that introduced a new term ‘t’. Replace ‘t’ in T with an appropriate variable to form an open sentence R`. Lewis now claims that ‘t’ is correctly defined as follow:

t = the unique x such that R`

Note the uniqueness requirement. If there are multiple realizations of R` (that is, variable assignments that satisfy R`) differing in what they assign to x, then ‘t’ is denotationless. **Van Fraassen (1997) argues that, provided only that T is consistent and has an infinite model, such will always be the case.**

Let R be the existential closure of R`–that is to say, the Ramsey sentence of T.

(1) Assuming consistency throughout, there will be a model in which R` is true on some assignment of values of the variables.

(2) If T has an infinite model, then R` has many non-isomorphic such realizations, each of which yields a model in which R is true.

From these it follows that if T is consistent and has an infinite model, then Lewis’ proposed uniqueness postulate is false. The model theoretic facts appealed to here are basically those used in Putnam’s model-theoretic argument—in this sense, Putnam’s argument could be taken as a direct objection to Lewis’ theory of science. [30]

Sure, if T is consistent and has an infinite model, then R` has many non-isomorphic realizations, each of which yields *a* model in which R is true. Does it follow that Lewis’ uniqueness postulate is false? No, in order to show that, we would need to show that R` has multiple realizations, each of which yields an* intended *model of R. What counts as an intended model of R here? Answer: one that assigns the *previously fixed *extensions to the vocabulary that was around prior to the theory T—in Lewis’ terminology, the O-vocabulary. Lewis reminds us several times that what he means by ‘realization’ is a realization that yields a model that is standard with respect to the O-vocabulary.

Finally, I should say again that we are talking only about realizations that make T true under a fixed interpretation of all of its 0-vocabulary…

John Winnie has announced a proof that scientific theories cannot be uniquely realized. Though his proof is sound, it goes against nothing I want to say. Most of Winnie’s multiple realizations of a given theory-all but one, perhaps-are not what I call realizations of the theory. I am concerned only with realizations under a fixed interpretation of the 0-vocabulary. [1970: 433-434]

Here’s a simple example. Let T be the sentence ‘There is a unique thing that is H and it is G and there are infinitely many things that are F.’ Thus, T is consistent and has only infinite models. Now suppose that ‘H’ is the newly introduced term and that the previously fixed extensions of ‘G’ and ‘F’ are {o1} and {o1, o2, o3…} respectively. Although there are many realizations simplicter of ‘There is a unique thing that is Phi and it is G and there are infinitely many things that are F’ only one of these realizations is a realization in Lewis’ sense–that is, a realization that yields models that assign {o1} and {o1, o2, 03,…} to ‘G’ and ‘F’ respectively. In particular, only the realization that assigns {o1} to ‘Phi’ is a realization in Lewis’ sense. So here we have a theory T that is both consistent and has an infinite model, and yet there is a *unique *realization of R`, in the sense of ‘realization’ that Lewis has in mind (i.e., realization that yields an intended model).

I think Lewis was right: Putnam’s model-theoretic argument is a successful reductio on global descriptivism, but it can’t touch the (admittedly modest) local descriptivism of “How to Define Theoretical Terms”.