Suppose you pursue truth for its own sake. As we learn from Aristotle, it does not follow that you don’t pursue truth for the sake of something else. For the most valuable things are both intrinsically and instrumentally valuable, and so they are typically pursued both for their own sake and for the sake of something else.
What if you pursue something, but not for the sake of something else. Does it follow that you pursue the thing for its own sake? Maybe, but it’s not as clear as it might seem. Imagine that you eat fiber for the sake of preventing colon cancer. Then you hear a study that says that fiber doesn’t prevent colon cancer. But you continue to eat fiber, out of a kind of volitional inertia, without any reason to do so. Then you are pursuing the consumption of fiber not for the sake of anything else. But merely losing the instrumental reason for eating fiber doesn’t give you a non-instrumentally reason. Rather, you are now eating fiber irrationally, for no reason.
Perhaps it is impossible to do something for no reason. But even if it is impossible to do something for no reason, it is incorrect to define pursuing something for its own sake as pursuing it not for the sake of something else. For that you pursue something for its own sake states something positive about your pursuit, while that you don’t pursue it for the sake of anything else states something negative about your pursuit. There is a kind of valuing of the thing for its own sake that is needed to pursue the thing for its own sake.
It is tempting to say that you pursue a thing for its own sake provided that you pursue it because of the intrinsic value you take it to have. But that, too, is incorrect. For suppose that a rich benefactor tells you that they will give you a ton of money if you gain something of intrinsic value today. You know that truth is valuable for its own sake, so you find out something. In doing so, you find out the truth because the truth is intrinsically valuable. But your pursuit of that truth is entirely instrumental, despite your reason being the intrinsic value.
Hence, to pursue a thing for its own sake is not the same as to pursue it because it has intrinsic value. Nor is it to pursue it not for the sake of something else.
I suspect that pursuing a thing for its own sake is a primitive concept.