One thing that characterizes agency is the ability to act intentionally, as opposed to having things happen to you. Unsurprisingly, therefore, the difficulty about the constitution of agency is reflected in a puzzle about the intentionality of self-examination.
On the one hand, to self-examine is to perform an action—or better, to engage in an activity—intentionally. It cannot just be something that happened to us while we were not paying attention—at least not in the sense that our blood-pressure may drop without us noticing. For there to be self-examination in the first place, someone has to be the agent of the examination. In this sense, we have to be a party to what is going on. However, on the other hand, the intentionality of self-examination is of a special kind. To self-examine is not something we can simply perform as we would an action like shutting our eyelids. We cannot just obey the order: “Examine yourself!” (“Know thyself!”). Obeying this order necessarily involves figuring out first what it would take to obey it, and essentially this is not given.
The problem of intending to engage in self-examination may be compared to the problem of talking of the next prime after 37 in the series of prime numbers: Even though we have a strong sense that there must be such a series, we don’t have it; we only have a promise of one. Therefore we don’t have what it takes to refer to the series, and all our references to “it” rely on our sense that there has to be one, and the promise that there is. Likewise, we may intentionally examine ourselves only to an extent: we may only act under the guidance of a promise that we are really doing something—a sense that we must be. Whether we are or not is, while we are engaged in this would-be self-examination, remains to be seen.
We may also come to realize that we are self-examining. That is, the fact that there is a riddle about ourselves in the first place may be revealed to us for the first time when the riddle begins to be solved. This is the sense in which self-examination may happen to us while we don’t notice. In this sense self-examination is an activity of which we may discover ourselves to be the agent.