In his introduction, Kai von Fintel tells an anecdote that I think sums up why it is sometimes difficult to explain what it is we do. Some time ago, Emmon Bach wrote a book for Cambridge University Press on if/then conditionals. The copy-editor sent it back, replacing every use of “if and only if” with a simple “if,” saying the “and only if” was redundant.
As it turns out, although people often interpret “if” as meaning “if nd only if,” that’s simply not what the word means, despite our intuitions (most people interpret if you mow the lawn, I’ll give you $5 as meaning if and only if you mow the lawn…).
Part of the mystery, then, is explaining why our intuitions are off. In the meantime, though, explaining what I do sometimes comes across as trying to prove the sky is blue.