If we can only translate declarative sentences into symbolic logic, then how is symbolic logic useful for linguistics?

Question

I can see how SL can be a decent metalanguage for doing cross-linguistic semantic work, but I feel like it's severely limited by the fact that you can't translate any kind of non-declarative sentence. This is means that only tiny fraction of any one given language can ever be logically represented. How is symbolic logic useful in linguistics?

Answer 1

David Lewis' account of the logic of imperatives is in terms of the possible worlds in which the imperative is obeyed. Here is a handout for a class which extensively deals with the formal semantics of questions.

Anyhow, there is more to what linguists do than assign semantic interpretations to sentences. We use formalisms for all aspects of language, and at least for the non-quantitative, the interpretation of those formalisms requires the use of logic. Formal logic is important in interpreting these formalisms, since it provides a clear method of interpretation. Modal logic has proven useful in accounting for the meaning of sentences, and is of no use in interpreting phonological rules. Not every useful tool have to be useful for all problems.

Answer 2

Logic concerns inference. Can you infer questions? Can you infer imperatives? Perhaps, but if so, by introducing assumptions about the nature of human conversation. I don't see why an extension of logic principles or representations would be necessary. You can represent nondeclaratives, if you want, in a notation resembling that of predicate logic, but it's not clear that these would count as "logical representations".

There are some proposals about representation of questions in McCawley's The Syntactic Phenomena of English, which help to make clear some subtle matters of scope.