Coarse-grained meanings and impossible worlds
To some extent, one can account for semantic phenomena without assigning meanings to words or sentences or thoughts. For instance, we might say that beliefs and other attitudes are relations to sentences, i.e. to strings of symbols. Roughly, to believe a sentence S is to be disposed to utter (or assent to) S (or some translation of S) under certain conditions. When people talk to each other, such dispositions may be transferred: after hearing me utter the sounds "it is raining", you acquire the disposition to utter those sounds yourself. Apart from communication, we can also account for things like synonymy and analyticity. Roughly, two sentences are synonymous if necessarily, anyone who stands in the belief relation to one of them also stands in the belief relation to the other. There is no compositional semantics in this picture, because there is no semantics at all. But there might be recursive rules for translating from one language to another.
What's missing in this picture are any relations between language and the non-linguistic world. Why does it matter to me whether you are disposed to utter the sounds "it is raining" rather than, say, "mitis laining", or "it isn't raining"? Why do we sanction the utterance of certain sounds and praise the utterance of others? How do people manage to get into a cinema or a plane by uttering certain sounds? What is the difference between a competent speaker of a language and someone who merely parrots everything they hear? A good answer to questions like these requires some link between linguistic expressions and non-linguistic matters.
At a minimum, we might draw a distinction between sentences that are true and sentences that are false. Equivalently, we might assign to every sentence one of two non-linguistic meanings, "the true" and "the false". We could then do a bit of compositional semantics with these meanings: we could give general rules relating the truth-values of certain complex sentences ("A or B") to the meanings of their parts. And we could say that there's a norm to utter only sentences that are true (or known to be true), which explains why it matters whether I say "it is raining" or "it isn't raining", and why it's not trivial to learn a language.
Of course, belief can hardly be just a relation to a truth-value, nor are truth-values the only things that are communicated from speaker to hearer. So here we might appeal once again to sentences: what is communicated is a truth-value under a particular sentential guise.
Even better, we might use more fine-grained entities, say, sets of possible worlds, as sentence meanings. Equivalently, we might distinguish sentences not only by their truth-value, but by their truth-conditions. If we're lucky, we could then also give a compositional semantics for some intensional constructions. Belief could be understood as a relation to a set of possible worlds under a sentential guise. Very roughly, to believe a set p under guise S is to (a) act in a way that would be reasonable if the actual situation were in p, and to (b) be disposed to utter (or assent to) S under certain conditions. Now we can explain why I can bring you to fetch an umbrella, or stay inside, by uttering the sounds "it is raining". And we can explain why it's useful to have linguistic conventions.
Perhaps at some point one or two people thought that one can account for all semantic phenomena in terms of truth-conditions, or sets of possible worlds. But, setting aside tendentious definitions of "semantic phenomena", it's obvious that this won't do, because sets of possible worlds are rather coarse-grained (as are, in a different way, Russellian propositions and other good candidates for semantic values). It makes a difference whether I say "17*21=357" or "if it's raining, then it's raining": the first might be helpful when the second is not. And there are plenty of linguistic constructions where the possible-worlds meaning of complex sentences is not determined by the possible-worlds meaning of their parts.
So why not go one step further and use very fine-grained meanings, so that we can account for all semantic phenomena in terms of meanings? Let S1 and S2 have different meaning whenever there is an embedding X (e.g. "Jones proved that ...") such that X(S1) is true and X(S2) is false, and whenever different communicative roles can be performed by uttering S1 and S2. This would mean that any two sentences whatsoever would have different semantic values. A simple and popular way to provide such fine-grained meanings is to define them as sets of possible or impossible worlds, where the "worlds" are simply sets of sentences (or some other entities from which such a set can be read off). For example, the semantic value of "it is raining" is the set of sets of sentences containing the sentence "it is raining". We can then say that belief is only a relation to a set of (possible or impossible) "worlds", without mentioning any sentential guise, and we can specify the meaning of every sentence, no matter how hyperintensional, entirely in terms of the meaning of its parts.
But now we've come full circle. The alleged semantic values in the final account are really just the sentences themselves, decorated with set-theoretic distractions. The difference between a sentence S and the set of sets of sentences containing S is much like the difference between S and the unit set of S: there is a trivial 1-1 map between the two kinds of thing. Any theory involving the one can trivially be converted into a theory involving the other. For example, have a look at the compositional semantics with possible and impossible "worlds". In ordinary possible world semantics, the meaning of "it is not raining" is the complement of the meaning of "it is raining": a possible world is in the semantic value of the first sentence iff it is not in the value of the second. Not so if we have impossible "worlds" where it is both raining and not raining. In the impossible-worlds framework, we don't need to look at the meanings of the parts of a sentence at all. We know from the start that the meaning of "it is not raining" is precisely the set of sets of sentences containing the sentence "it is not raining". Compositional semantics has become utterly trivial.
Semantics with very fine-grained (hyperintensional) semantic values is really the same thing as semantics without semantic values. As before, we thereby lose every contact to the non-linguistic world. Elementary set theory is enough to know the hyperintensional impossible-worlds content of every French sentence. But this won't help you in the streets of Paris. If you only know that "il pleut" means the set of sets of sentences containing "il pleut", you will have no idea whether to pack an umbrella or sunscreen when people utter those words. You won't actually know what the sentence means.
The lesson is that we shouldn't be over-ambitious. Let's have coarse-grained semantic values that provide us with non-trivial language-world relations. And let's concede that not all semantic phenomena can be accounted for on this level. Sometimes, for example, the meaning of a complex sentence will be determined not only by the meaning of its parts, but also by the sound and shape of its parts. This is not a problem. Those non-semantic features are after all not a secret to competent speakers. If you know a language, you will trivially know the hyperintensional impossible-worlds meanings in addition to the possible-worlds meanings, so we shouldn't be surprised if the possible-worlds meaning of some complex constructions also makes use of impossible-worlds meanings.
So we do need hyperintensional, impossible-worlds meanings. I'd rather we call them by their right name: they are simply the sentences themselves, individuated as strings of sounds and shapes. But we should not let these things replace our coarse-grained meanings.
Interesting stuff, but it may be worth pointing out (maybe too obvious) that sounds and shapes won't do it alone - otherwise 'if every woman likes a man then every woman likes a man' which has one trivial meaning, one non trivial meaning but the same shapes and sounds will be problematic. Same with 'they saw her duck but not her duck'.