A great puzzle of twentieth-century philosophy of language was, how are finite beings able to understand a potential infinity of sentences? The answer is supposed to be that understanding is recursive: infinitely many sentences can be constructed out of finitely many words combined according to finitely many rules; we understand a sentence by understanding the words in it and knowing the relevant rules. A great puzzle of twenty-first-century philosophy of language is shaping up to be this: how do we reconcile the solution to the previous puzzle with what sentences actually strike us as saying? It's a puzzle because S's compositionally determined meaning is not always a very good guide to what S intuitively says, or to its contribution to what is said by sentences in which S is embedded. This chapter focuses on the more radical case where a sentence says something its meaning positively disallows, such as the case where a sentence's real content is not a possible semantic content.