You suggest that what distinguishes the beliefs commonly criticized as "belief in belief" from others is that they are voluntarily, explicitly adopted (and that this goes along with treating them in all-or-nothing fashion rather than as having degrees of assent). I think there's a lot of truth in this. But your taxonomy of beliefs seems to be missing a category. You say:
The answer to this is that choosing to believe something is more than choosing to say it to oneself and to others. Rather, it is choosing to conform the whole of one's life to the idea that this is true.
But a large part of the point of talking about "belief in belief" is that some people say they believe things (voluntarily, black-and-white-ly) but don't act in ways fully consistent with believing them. For instance, some (a small minority, and I'm not intending to insinuate e.g. that most religious people are like this) profess to believe that in the very near future the Second Coming of Christ will occur, putting an end to pretty much all the institutions of present-day human society -- but they aren't taking out enormous loans that will come due after the alleged Second Coming, they are still sending their children to school, they still have pensions, etc., etc., etc. Which suggests that maybe they don't really believe that the world as we know it is going to end in a few years, they just think they're supposed to say they do. (Very similar things are sometimes said about singularitarians. Again, I am not claiming that they are correctly said of more than a small minority.)
In other words, the point is that someone who says "The Messiah is coming" is not
in every way acting as though this is true
and indeed this is how we might diagnose that
his internal Bayesian estimate is that [...] the evidence is strongly against this
and declare him a practitioner of "belief in belief". Unknowns, consider the religious people in the second category you mention in your last paragraph. How do you know that they
clearly have an internal estimate which is extremely low: they are virtually certain that their religion is false
if, as you say,
in every way, externally and internally, they act and think as though it were true?
Ok, you are right that I missed the situation of people who say things but fail to live up to them in obvious ways. That is probably simply because I don't really see this in the religious people I am most acquainted with. My guess is that if we leave out people who are actually not telling the truth about what they believe (i.e. people who are literally lying about what they hold), that happens mainly because of akrasia and other similar situations where their desires are in conflict with their explicit beliefs about what they should do, and they follow t...
I am not sure that it is productive to tell certain people that they do not really believe what they claim to believe, and that they only believe they believe it. I have an alternative suggestion that could possibly be more useful.
Binary Beliefs
It seems that human beings have two kinds of beliefs: binary beliefs and quasi-Bayesian beliefs. The binary beliefs are what we usually think of as beliefs, simple statements which are true or false like "Two and two make four," "The sun will rise tomorrow," "The Messiah is coming," and so on.
Binary beliefs are basically voluntary. We can choose such beliefs much as we can choose to lift our arms and legs. If I say "the sun will rise tomorrow," I am choosing to say this, just as I can choose to lift my arm. I can even choose the internal factor. I can choose to say to myself, "the sun will rise tomorrow." And I can also choose to say that the sun will NOT rise. I can choose to say this to others, and I can even choose to say it to myself, within my own head.
Of course, it would be reasonable to respond to this by saying that this does not mean that someone can choose to believe that the sun will not rise. Even if he says this to himself, he still does not act as though the sun is not going to rise. He won't start making preparations for a freezing world, for example. The answer to this is that choosing to believe something is more than choosing to say it to oneself and to others. Rather, it is choosing to conform the whole of one's life to the idea that this is true. And someone could indeed choose to believe that the sun will not rise in this sense, if he thought he had a reason to do so. If he did so choose, he would indeed begin to make preparations for a dark world, because he would be choosing to conform his actions to that opinion. And he would do this voluntarily, just as someone can voluntarily lift his arm.
Quasi-Bayesian Beliefs
At the same time, human beings have quasi-Bayesian beliefs. These are true degrees of belief like probabilities, never really becoming absolutely certain of the truth or falsity of anything, but sometimes coming very close. These are internal estimates of the mind, and are basically non-voluntary. Instead of depending on choice, they actually depend on evidence, although they are influenced by other factors as well. A person cannot choose to increase or decrease this estimate, although he can go and look for evidence. On account of the flawed nature of the mind, if someone only looks for confirming evidence and ignores disconfirming evidence, this estimate in principle can go very high even when the objective state of the evidence does not justify this.
Belief in Belief
It seems to me that what we usually call belief in belief basically means that someone holds a binary belief together with a quasi-Bayesian belief which conflicts with it. So he says "The Messiah is coming," saying it to himself and others, and in every way acting as though this is true, even though his internal Bayesian estimate is that after all these thousands of years, the evidence is strongly against this. So he has a positive binary belief while having a very low estimate of the probability of this belief.
The reason why this often happens with religion in particular is that religious beliefs very often do not have huge negative consequences if they are mistaken. In principle, someone can choose to believe that if he jumps from the window of the tenth story of a building, he will be ok. In practice, no one will choose this on account of his non-voluntary Bayesian estimate that he is very likely to be hurt if does so. But a person does not notice much harm from believing the Messiah is coming, and so he can choose to believe it even if his internal estimate says that it is likely to be false.
A cautionary note: one might be tempted to think that religious people in general have belief in belief in this sense, that they all really know that their religions are unlikely to be true. This is not the case. There are plenty of ways to distort the internal estimate, even though one cannot directly choose this estimate. I know many very religious people who clearly have an extremely high internal estimate of the truth of their religion. They REALLY BELIEVE it is true, in the fullest possible sense. But on the other hand I also know others, also extremely devout, who clearly have an internal estimate which is extremely low: they are virtually certain that their religion is false, and yet in every way, externally and internally, they act and think as though it were true.