One of the few authors who has attempted to address the question of what it is about the linguistic behaviour of a population which justifies our saying that some particular model-theoretic interpretation is the right one for the language they speak is David Lewis (1975). Since I have already used ℒ for a syntactically specified language I will use ℒ+ for a language together with its interpretation rather than Lewis's ℒ. Lewis's proposal is (Lewis 1975, p. 7)

the convention whereby a population P uses a language ℒ+ is a convention of truthfulness and trust in ℒ+. To be truthful in if is to act in a certain way: to try never to utter any sentences of ℒ+ that are not true in ℒ+. Thus it is to avoid uttering any sentence of ℒ+ unless one believes it to be true in ℒ+. To be trusting in ℒ+ is to form beliefs in a certain way: to impute truthfulness in ℒ+ to others, and thus to tend to respond to another's utterance of any sentence of ℒ+ by coming to believe that the uttered sentence is true in ℒ+.

Where Lewis speaks of an interpreted language, which I have represented by ℒ+ in quoting him, I prefer to speak of a language if, together with an interpretation (W,D,V). I shall assume that speakers of ℒ+ are members of D and that their properties and the relations among themselves and to other things are all appropriate set-theoretical entities constructed out of W and D. In particular, their beliefs may be construed as relations to propositions.