Very many people have pointed out to me that this initial worry is just a terminological confusion. The obvious solution to my puzzle is to say that it equivocates on the idea that cognitive control leads to the acquisition of new beliefs. This is ambiguous, because this idea could either be spelled out as the thought that cognitive control is the belief acquisition event, or as the idea that cognitive control is the process that leads up to that event. Put like that, the puzzle seems to go away, because the whole debate around doxastic voluntarism is concerned with the event, while cognitive control intuitively is about the cognitive process leading up to it.
This is where the last of the themes of the book comes in. I agree that it is possible to solve the puzzle this way, but I hold that going for that solution has a consequence that is at least surprising, but for many people presumably also rather unappealing. I argue that if cognitive control can be voluntary and cognition at the same time, then that entails that extended cognition must be true.
To see why this is so, I turn to an argument by Yair Levy. Levy has recently rightly pointed out that there is no relevant functional difference between voluntary epistemic actions inside the head and voluntary epistemic actions that involve the environment, like calculating on paper. If this is right (and it seems right to me) then it forces a surprising choice.
Given Levy’s argument, if cognitive control consists in voluntary epistemic actions and if cognitive control is supposed to be cognition then this must be so, whether or not the cognitive control action includes the environment. If on the other hand one wants to deny that extended cognition is true, then this implies that voluntary cognitive control is not cognition.
Once the epistemic agency in cognitive control has been clarified in this way many surprising things follow, and this is what the rest of the book is for. It explores the surprising consequences for topics from dual processing (where it suggests that judgements cannot be system two if extended cognition is not true) to moral psychology where it provides an argument for extended willpower.
More generally, the book uses the distinction between the two ways to connect epistemic agency and cognition to make some progress on the age old question of what distinctly human cognition might consist in.