In Ch. 3 of Making It Explicit, Robert Brandom asks a very interesting question. He asks what it is that makes a mapping from one set to another a specifically semantic interpretation of something. For Tarski’s explanation of quantifiers in terms of topological closures, he says that it is because it defines a notion of logical consequence that is appropriate to the idiom that Tarski is using. So is it defining logical consequence that is necessary for something to be a semantic interpretation? No. He says that it is taking thing as sentences which means treating them as susceptible to the rules of reason, evaluation in terms of truth, and usable in derivations in new inferences. He goes on to say that formal semantics is semantics only if it presupposes being hooked up to some kind of appropriate pragmatics. He contrasts this picture with what Lewis does in “Language and Languages” and what Stalnaker does in Inquiry. I’m not clear on the details of the contrast, so I will come back to that later.