Coded communication
Imagine a community of people who pass encrypted messages to one another, without knowing what they mean. Agent X has encrypted a message and handed it to messenger A, who passes it to messenger B, who passes it to agent Y, who has the codebook to decrypt the message. When A utters the message to B, she has no idea what it says; neither does B.
Intuitively, the meaning or content of A's utterance is the content of the decrypted message. That's why A and B don't know what the utterance means.
This might become less obvious if the code has the grammatical form of an ordinary sentence. Suppose it is "Heynya is noor." It's easy to imagine that messengers like A and B start to embed these codes in other sentences. A might ask "Have I already told you that Heynya is noor?" B might ask, "Do you know if Heynya is noor?", and A might answer, "I do". They might say, "'Heynya is noor' is true iff Heynya is noor". If they learn to say such things, someone might claim that they do know what the codes mean. They know the relevant T-theory, for example.
To see that they still don't really know what the codes mean, we have to look beyond their verbal behaviour. Suppose the decrypted meaning of "Heynya is noor" is that the Ashka'ri tribe is planning an attack. If B learned that the Ashka'ri tribe was planning an attack, he would take precautions. His "belief that Heynya is noor" has no such effects.
We might assign a different kind of meaning to the codes, a meaning that is genuinely known to the messengers. What B learns when A utters "Heynya is noor" is that whatever the codebook assigns to "Heynya is noor" is the case. This, we might say, is the actual meaning of the coded sentence among the population of messengers. (Here I assume that the messengers have reason to believe that the coded messages are generally accurate.)
We might call this known type of meaning the primary intension of a code; the secondary intension is given by the codebook: it is the code's "original" meaning, the one that isn't genuinely known among the messengers.
At this stage, primary intensions are fairly uninformative. We can change this by giving the messengers partial knowledge of the codebook. More simply, let's assume that the code uses some words in their ordinary meaning: it only encrypts singular terms. The message passed by A to B might now say "Heynya is planning an attack", where "Heynya" is the only encrypted part, and this is known among the messengers.
As before, B doesn't know the original, secondary meaning of the message. But now he acquires useful information. For one, he learns that someone is planning an attack, which may be sufficient to take precautions. He also learns that whoever composed the original message knows that this party is planning an attack. And he learns that whatever earlier messages said about "Heynya" applies to the party planning an attack. For example, if an earlier message went "Heynya is powerful", he can now infer that a powerful party is planning an attack.
In this setting, the primary intension of "Heynya is planning an attack" is the proposition that whoever the codebook assigns to "Heynya" is planning an attack. It has become much more informative.
A grammar that maps every message to its primary intension can play an important role in a systematic theory of the messengers' practice. For example, we might want a grammar to deliver meanings that fill the blank in: when someone hears an utterance of S, they tend to act in a way that would be reasonable on the assumption that —. Primary intensions can fill this blank, secondary intensions can't.
On the other hand, the primary intensions probably won't fit how the messengers talk about their own language. If we ask B what he learned from A, he might well answer "that Heynya is planning an attack", not "that whoever the codebook assigns to 'Heynya' is planning an attack". So the primary intensions may not match inner-language judgements about "what is said".
Also, the primary intensions are still uniform for all singular terms. All singular terms have the same kind of primary intension: "whatever the codebook assigns to '…'". There's no variation. So a grammar doesn't really need a separate entry for each singular term. We can instead have a kind of meta-rule that fills in all these entries at once.
In fact, we can reconstruct the "primary grammar", the grammar that maps expressions to their primary intensions, from the "secondary grammar" that maps expressions to their secondary intensions. Suppose, as before, that the codebook maps "Heynya" to the Ashka'ri tribe. So the secondary intension of "Heynya" rigidly picks out the Ashka'ri, and the secondary intension of "Heynya is planning an attack" is the proposition that the Ashka'ri tribe is planning an attack. What's the primary intension? Well, it replaces the rigid secondary intension of "Heynya" by its non-rigid primary intension, and we know what that is, since it's essentially the same for all singular terms.
So while the primary intensions can play a useful role, it is understandable that people studying the language of the messenger population would focus on the secondary grammar.
In some respects, English is a messenger language. Technical terms like 'boson' are widely used to mean "whatever the experts say they mean"; the experts have the codebook. Proper names like "Gödel" are widely used to mean "whoever stands at the origin of our use of this name"; the causal chain is the codebook.
I think this explains why people studying English tend to focus on the secondary grammar, identifying the meaning of "Gödel" with Gödel and ignoring primary intensions.
Having learned to think about meaning from Frege and Lewis, I always found this puzzling. Why, I thought, should we focus on a notion of meaning that makes meanings opaque to competent speakers and hearers? Shouldn't we rather identify meanings with something that is actually, transparently conveyed from speaker to hearer?
Well, I still think we should, but the reason is more subtle than I used to think. English only works approximately like the messenger language. "Jack the Ripper" is a proper name, but its primary intension isn't just "whoever stands at the origin of our use of this name". "Hesperus" and "Phosphorus" have substantively different primary intensions. "Boson" and "elm" certainly do. The primary intensions of names and kind terms in English aren't uniform, and they aren't determined by their secondary intensions. So something gets lost when we focus on secondary intensions.