As was argued in an earlier post, any belief, or at any rate almost any belief, is voluntary insofar as we choose to think, act, and speak as though it were true, and in theory it is always in our power to choose to behave in the opposite way. In practice of course we would not do this unless we had some motive to do it, just as the fact that it is in someone’s power to commit suicide does not make him do so before he has a motive for this.
Such a belief, since it involves affirmation or denial, has a basically binary character — thus I say either that the sun will rise tomorrow, or that it will not. This binary character can of course be somewhat modified by the explicit addition of various qualifiers, as when I say that “I will probably be alive five years from now”, or that “there is a 75% chance that Mike will come to visit next week” or the like. Nonetheless, even these statements have a binary character even if at one remove from the original statement. Thus I can say “Mike will come to visit,” or “Mike will not come to visit”, but also “there is a 75% chance etc” or “there is not a 75% chance etc”.
The interior apprehension of the mind, however, does not have the same binary character, but is more a matter of degree. Thus for example someone may argue that increasing the restrictions on gun ownership in the United States would be a good idea. “More gun control would be good,” he says. This is the affirmation of one side of a contradiction. Then suppose he is involved in a conversation on the matter, with another person arguing against his position. As the conversation goes on, he may continue to assert the same side of the contradiction, but he may grow somewhat doubtful inside. As he walks away from the conversation, he still believes that more gun control would be good. He still chooses to speak, think, and act in that way. But he is less convinced than he was at first. In this sense his interior apprehension has degrees in a way in which the belief considered as an affirmation or denial does not.
Probability theory is a mathematical formalization of such degrees of belief as interior apprehension, and the laws of probability are rules which must be followed in order to maintain logical consistency when working with such degrees. In another post we looked at Hume’s claim about the impossibility of induction. It is likely that one cause of his error was the fact that the formal theory of probability was less developed at the time; in particular, Bayes’ theorem, which we used against Hume, was proved 20 years after Hume’s treatise was written.
Thus, in our speech and behavior, beliefs are basically binary, but we possess various degrees of certainty about our beliefs. And such a degree is reasonably considered to be something like the probability, as far as we are concerned, that our belief is true.