Brentano taught us to ask after the reference of a mental state: mental states are always “about” something, “directed at” something. They are not sealed off from the world but engaged with it. They are relational not intrinsic. This intentionality distinguishes them from merely physical states, which are not about things. But did he go far enough in discerning mental aboutness? What about the relation a mental state has to the subject who possesses that state: isn’t the mental state “directed” to its subject also? Doesn’t it “point” inward as well as outward? Can we not say that every mental state stands in the relation of being had by a subject as well as the relation of being about an object? There is a kind of double intentionality at work.
Is there any further type of intentionality? What about the idea that all mental states have a kind of self-awareness built into them? We are always aware of our awareness. This need not be conscious awareness; it is sometimes called “pre-reflective”. At the least a mental state is always potentially an object of awareness—just a short step away. So the mental state is always about an object, had by a subject, and cognizant of itself (for-itself, as Sartre would say). It points in three directions simultaneously: at its subject, at its object, and at itself. Is that it?
Consider belief or thought: here we have the triple intentionality just mentioned—the subject, the object, the mental state itself. But isn’t there also a fourth layer, viz. the proposition believed? The subject stands in the belief relation to that proposition—it is the content of his belief. So he has the belief, the belief is in a certain proposition, the proposition is about a certain object, and he is aware of having that belief. Now we have four types of mental directedness—quadruple intentionality. Brentano needed to add an additional three levels of directedness to his simple one level account. The mind points in many directions at once.
Can we generalize this point about belief? Take grasping a sense: there is the subject doing the grasping, there is the object (reference) of the grasping, there is the sense that is grasped, and there is the grasping itself. So understanding a word can be said to involve all four mental relations: having the understanding, apprehending a meaning, referring to an object, and being aware of itself. All are necessary to linguistic understanding. The same structure applies to perceptual states: the subject undergoes a perceptual experience, which is of an aspect of the external world, which presents an object, which is itself apprehended (albeit implicitly). So the experience “points” at four different things: a subject, an object, an aspect, and itself. For example, I see a table under the aspect brown, shiny, etc., and my seeing a table is presented to me. The experience sits at the center of a web of connections with other things (including itself): it is multiply ostensive. It points at many things. Frege had already introduced another layer of intentionality over and above Brentano’s “aboutness” by suggesting that we grasp senses as well as refer to objects, but we can go further and recognize the involvement of the subject (self) and the mental state itself—these are ostensive objects too.  A word has a sense, a reference, a subject who understands it, and that understanding itself (when the word is understood). If I understand a name, say, I grasp its sense and denote its reference, but I also have that understanding and am aware of my understanding. I point in four different directions simultaneously (or my understanding does). There is not just one relation–as it might be, denoting–but several relations, which are all integral to the state of understanding. Intentionality proliferates. The mind is a multi-directional thing.
 Actually Frege anticipated this enrichment to his theory of sense and reference in “The Thought” by insisting that every mental state has a subject. That implies that the mental state of grasping a sense necessarily brings with it another object, viz. the self or subject. There is no such thing as a free-floating grasp of a sense; all such grasping is by a subject. So, according to Frege, sense inherently points both to an object of reference and to a subject who apprehends the sense. Indeed, that subject must necessarily exist, according to Frege, while the object of reference need not exist (Pegasus etc.). Sense is precisely something that mediates between subject and object. The ontology of sense therefore incorporates an ontology of subjects and objects.