(1) direct activism: we can form beliefs directly
(2) weak voluntarism: we can form beliefs directly for pragmatic reasons
(3) strong voluntarism: we can form beliefs directly for pragmatic reasons and without epistemic support
(All of these would be classified as 'doxastic voluntarism' by most people.) Frankish suggests that (1) tends not to be separated out because there is a plausible case that it implies (2); but that it does not imply (3). Strong voluntarism he regards as necessarily false, but weak voluntarism as very defensible. The case against strong doxastic voluntarism is interesting and worth looking at.
Frankish begins with a principle, what he calls the Revised Williams Principle:
For any proposition p, it is impossible to believe in full consciousness that one consciously believes that p and that one's belief that p is both unsupported and deviant.
The basic idea, crudely put and without worrying too much about precise nuances, is that we can't believe something if we think that there is no reason to believe it (unsupported) and that the process of coming to believe it is not connected with the truth (deviant). The 'both' is important; Frankish thinks there are marginal cases where we probably can believe things we think are unsupported as long as we think they are nondeviant, and where we probably can believe things we know to be deviant if we think they are supported.
Now, let's say that we are being wanton if we believe ('in full consciousness') that we believe something on purely pragmatic reasons and have no evidence to believe it true. Strong voluntarists hold that we can do this. But, Frankish says, this violates the Revised Williams Principle: this would make beliefs accepted wantonly both unsupported and deviant, at least for typical cases. But this deals with coming to believe rather than believing. There are two kinds of change that could go on simultaneously with the possibly make the situation consistent with the Revised Williams Principle: (1) I come to believe that it is supported; (2) I forget that it is deviant. But, if we are talking about 'full consciousness', the belief in (1) runs into the same problem: it is unsupported and deviant. And we cannot simply forget things the way (2) requires -- Frankish, without committing to an explanation of this, suggests that it could be due to an analogous situation -- to forget something at will we would also have to forget that we forgot it. Thus, the idea goes, strong voluntarism requires that both conditions are under our voluntary control; but they cannot be simultaneously under our voluntary control.
As Frankish notes, this argument requires that it all be done in full confidence -- obviously people do believe things that happen to be both unsupported and deviant. What they cannot do, according to the argument, is will to believe things that are both, which requires consciousness of both the lack of support and the deviance. And, of course, Frankish goes on to argue that you can have doxastic voluntarism that does not require this incoherent willing.
I think an issue of slippage in the actual course of the argument, even granting its principles, is that Frankish assumes that when we recognize that we are believing something only on pragmatic grounds that we recognize that our belief is deviant. There's good reason to think this false, I think. That is to say, I think it is generally the case that people regard even purely pragmatic reasons as broadly and loosely truth-conducive, in at least the sense that much of the time they tend to get you closer to the truth even in the absence of supporting evidence. Take, for instance, the idea that we should go with the simplest theory that fits the evidence. By the very set-up of the claim, 'simplicity' does not affect the evidence; it's a purely pragmatic reason for accepting a theory. But most people would regard simplicity as truth-tending in at least some limited way. Frankish would fit this, I imagine, into the category of 'thought to be unsupported but not deviant'. But what pragmatic reasons do we actually take to be deviant when we are actually using them? What does it mean to take something as a 'reason' in the first place? Perhaps little more than 'it may be relevant to getting us closer to the truth or something in its neighborhood'. That's very weak, but it seems to suffice to block any certainty that it is deviant.
When Frankish originally characterizes strong voluntarism, he characterizes it as the position that "we can form beliefs directly, for pragmatic reasons, and without epistemic support" (p. 527). But if all pragmatic reasons are taken to be presumptively nondeviant when we are using them, then strong voluntarism in this sense does not violate the Revised Williams Principle: consciously believing that one's belief is only for pragmatic reasons does not suffice to make it a case of consciously believing that one's belief is deviant.
Most people who advocate doxastic voluntarism do not, I think, much care whether it is weak or strong in Frankish's sense, and probably actually accept a weak voluntarism; there are only going to be occasional situations in which it would make a difference. It's rarely if ever going to be the case that you literally have no epistemic support whatsoever and yet have pragmatic reasons relevant to belief. (Pascal's Wager, for instance, the most famous argument suggesting doxastic voluntarism, requires only weak voluntarism, if that: the agnostic whom Pascal is addressing can perfectly well have reasons both for and against that leave him in suspense, and the Wager appeals to pragmatic reasons to break the deadlock.) But Frankish's argument, as it stands, slides too quickly between 'based only on pragmatic reasons' and 'deviant'.