According to him, there are at least two main options as to how exactly the notion of a precisification is to be cashed out:
“(1) One option is to construe a precisification of our vague language, L, as a precise language in its own right. (This is how David Lewis and Michael Dummett put it, at least in some of their works.) From this point of view, to say that L admits of several precisifications is to say that L is really many languages, a cluster of several (homophonic) precise languages whose semantics are only partially in agreement: our practices have simply failed to uniquely identify the one language that we are speaking. Correspondingly, to say that a statement of L is supertrue (for instance) is to say that it is true no matter how we suppose L tobe identified, i.e., no matter which (homophonic) variant of our statement we consider.
(2) A different, more popular option is to construe a precisification of a vague language L as a precise interpretation of L. (This is how most authors see it, from Kit Fine to the later David Lewis to Vann McGee, Brian McLaughlin, and Rosanna Keefe) Here the idea is that the grammar of our language is in principle compatible with countless interpretations, countless models each of which is logically adequate in that each assigns an extension to every predicate constant, a denotation to every individual constant, etc. Our linguistic practices and conventions are meant to select one such interpretation as the intended one, but they may fall short of doing the job properly. Correspondingly, to say that a statement of L is super-true (for instance) is to say that it is true no matter how we suppose the job to be done properly.”
As he himself anticipates, one might worry this to be a distinction without a difference. I do not find completely compelling the considerations he offers in response—and hope to post on it somewhen.
Here I just want to reckon that option (1) seems to me to be heterogeneous. In particular, one could adhere to a conception of precisifications as precise expressions (and languages) without necessarily reserving the expression ‘languages’ for precise ones—and thus in particular without being committed to the view of ordinary “languages” as clusters of (precise) languages.