In some of my published work, I have argued that:
(1.) Attitudes, such as belief and desire, are best understood as clusters of dispositions. For example, to believe that there is beer in the fridge is nothing more or less than to be disposed (all else being equal or normal) to go to the fridge if one wants a beer, to feel surprised if one were to open the fridge and find no beer, to conclude that the fridge isn't empty if that question becomes relevant, etc, etc. (See my essays here and here.)
(2.) Only conscious experiences are introspectible. I characterize introspection as "the dedication of central cognitive resources, or attention, to the task of arriving at a judgment about one's current, or very recently past, conscious experience, using or attempting to use some capacities that are unique to the first-person case... with the aim or intention that one's judgment reflect some relatively direct sensitivity to the target state" (2012, p. 42-43).
Now it also seems correct that (3.) dispositions, or clusters of dispositions, are not the same as conscious experiences. One can be disposed to have a certain conscious experience (e.g., disposed to experience a feeling of surprise if one were to see no beer), but dispositions and their manifestations are not metaphysically identical. Oscar can be disposed to experience surprise if he were to see an empty fridge, even if he never actually sees an empty fridge and so never actually experiences surprise.
From these three claims it follows that we cannot introspect attitudes such as belief and desire.
But it seems we can introspect them! Right now, I'm craving a sip of coffee. It seems like I am currently experiencing that desire in a directly introspectible way. Or suppose I'm thinking aloud, in inner speech, "X would be such a horrible president!" It seems like I can introspectively detect that belief, in all its passionate intensity, as it is occurs in my mind right now.
I don't want to deny this, exactly. Instead, let me define relatively strict versus permissive conceptions of the targets of introspection.
To warm up, consider a visual analogy: seeing an orange. There the orange is, on the table. You see it. But do you really see the whole orange? Speaking strictly, it might be better to say that you see the orange rind, or the part of the orange rind that is facing you, rather than the whole orange. Arguably, you infer or assume that it's not just an empty rind, that it has a backside, that it has a juicy interior -- and usually that's a safe enough assumption. It's reasonable to just say that you see the orange. In a relatively permissive sense, you see the whole orange; in a relatively strict sense you see only the facing part of the orange rind.
Another example: From my office window I see the fire burning downtown. Of course, I only see the smoke. Even if I were to see the flames, in the strictest sense perhaps the visible light emitted from flames is only a contingent manifestation of the combustion process that truly constitutes a fire. (Consider invisible methanol fires.) More permissively, I see the fire when I see the smoke. More strictly, I need to see the flames or maybe even (impossibly?) the combustion process itself.
Now consider psychological cases: In a relatively permissive sense, you see Sandra's anger. In a stricter sense, you see her scowling face. In a relatively permissive sense, you hear the shyness and social awkwardness in Shivani's voice. In a stricter sense you hear only her words and prosody.
To be clear: I do not mean to imply that a stricter understanding of the targets of perception is more accurate or better than a more permissive understanding. (Indeed, excessive strictness can collapse into absurdity: "No, officer, I didn't see the stop sign. Really, all I saw were patterns of light streaming through my vitreous humour!")
As anger can manifest in a scowl and as fire can manifest in smoke and visible flames, so also can attitudes manifest in conscious experience. The desire for coffee can manifest in a conscious experience that I would describe as an urge to take a sip; my attitude about X's candidacy can manifest in a momentary experience of inner speech. In such cases, we can say that the attitudes present a conscious face. If the conscious experience is distinctive enough to serve as an excellent sign of the real presence of the relevant dispositional structure constituting that attitude, then we can say that the attitude is (occurrently) conscious.
It is important to my view that the conscious face of an attitude is not tantamount to the attitude itself, even if they normally co-occur. If you have the conscious experience but not the underlying suite of relevant dispositions, you do not actually have the attitude. (Let's bracket the question of whether such cases are realistically psychologically possible.) Similarly, a scowl is not anger, smoke is not a fire, a rind is not an orange.
Speaking relatively permissively, then, one can introspect an attitude by introspecting its conscious face, much as I can see a whole orange by seeing the facing part of its rind and I can see a fire by seeing its smoke. I rely upon the fact that the conscious experience wouldn't be there unless the whole dispositional structure were there. If that reliance is justified and the attitude is really there, distinctively manifesting in that conscious experience, then I have successfully introspected it. The exact metaphysical relationship between the strictly conceived target and the permissively conceived target is different among the various cases -- part-whole for the orange, cause-effect for the fire, and disposition-manifestation for the attitude -- but the general strategy is the same.