Frege's mistake lay in assuming an appealing theory about why people say what they do. He thought people believe sentences true because they believe what those sentences say. If so, then what someone thinks must exactly reflect what he would candidly assert. The theory appeals because it assumes a even simpler picture of thinking: it is the picture of thinking as talking to yourself, sotto voce.
Frege didn't invent this way of thinking about thinking, but he was the first to see how much must follow about thought and language if it is true. If we take it as analytic that beliefs have the functional role Frege assumes for them, we can define Frege Beliefs:
x believesF p =df x would believe ┌p┐ was true iff x understood ┌p┐
When contemporary philosophers talk about the "narrow", "notional", "opaque" or "internalist" sense of 'believes' they are talking about beliefF.
The theory extends to the other attitudes. Thus we have Frege Desires:
x wantsF that p = df x would want ┌p┐ to be true iff x understood ┌p┐ .
And supposing that what you understand are meanings, we can define Frege Meanings :
x understands the meaningF of ┌p┐ =df x would believe ┌p┐ was true iff x believedF p.
Frege called the meaningsF of sentences "thoughts" and the meaningsF of words "senses". As Michael Dummett tells us: "The sense is that part of the meaning of an expression which is relevant to the determination of the truth‑value of a sentence in which the expression occurs". We can specify the Frege Meaning or sense of a term thus:
x understands the meaningF of ┌a┐ =df x would believe ┌..a...┐ was true iff x believedF ...a...
After Frege, when philosophers spoke about "meanings" (and after Frege, they spoke of little else) they were always talking about meaningsF.
These equivalences don't count as ampliative definitions since meaningF and beliefF are defined reciprocally. A proper analysis would have to define either 'beliefF' or 'meaningF' in terms that did not assume the other. However, despite generations of effort, finding a non-circular analysis has proved problematic. It is one of those philosophical problems that has grown capital letters: it is " The Problem of Intentionality".
" Intentionality has to do with the directedness or aboutness of mental states — the fact that, for example, one's thinking is of or about something."
"Intentionality is the power of minds to be about, to represent, or to stand for, things, properties and states of affairs."
This makes "intentionality" sound like some sort of spooky power. But on reflection there seems nothing spooky, or particularly mentalistic, about "aboutness". You can throw your arm about your friend. The moon circles about the earth. The price of gas is about four bucks a gallon. Nothing mental here. These kinds of "aboutness" are just garden variety relations that one item can have to another.
What is distinctive about the "aboutness" that is supposed to characterize mental states is precisely that it is not a relation between a mind and what it thinks about. Relations require relata. You can't put your arm about something unless it exists. On the other hand it seems you can think "about something" even if it doesn't exist. Don't children have beliefs about Santa? They must. After all, they say things like “Santa is coming tonight." Then, too, ordinary "aboutness" relations are relations between things, no matter what you call them. A moon couldn't orbit about Hesperus without orbiting about Phosphorus; Hesperus is Phosphorus. But apparently you can think about Hesperus without thinking about Phosphorus; after all, someone could think 'Hesperus has a moon' was true but deny 'Phosphorus has a moon'.
So what's special about the aboutness of intentional states is that these states manage to be "about " things without actually being... well, about things. Which, come to think of it, is kind of spooky.