Degrees all the way down: Beliefs, non-beliefs and disbeliefs

A Cartesian skeptic must not accept anything but what is ideally clear and distinct in her mind. She has only few beliefs, but all her beliefs have maximal certainty. Some philosophers recommend that a responsible believer should believe only what is beyond any doubt, namely logical and analytical truths. The common core of such proposals is that maximal certainty is held to be necessary and sufficient for rational belief. Consequently, the believer’s beliefs are all on an equal footing, no degrees of belief are needed, the set of beliefs is ‘flat’. Other philosophers felt that this picture is inadequate. As a matter of fact, we are not as reluctant to take on beliefs as Descartes admonished us to be. We believe quite a lot of things, and we are aware that there are differences in the quality of these beliefs. There is no denying that we are fallible, and if we are forced to give up some of the beliefs we have formed (Descartes would say, precipitately), we can adapt our beliefs in accordance with their varying credentials. Beliefs can be thought of as being equipped with labels specifying their ‘certainty’, or, to use a different terminology, their ‘entrenchment’ in a person’s belief state. Ever since Hintikka (1962), philosophers and logicians have been fond of thinking of belief as a form of necessity (‘doxastic necessity’). If one wants to acknowledge distinctions in degrees of belief, one has to introduce a notion of comparative necessity. Saying that A is more firmly believed than B is to say that A is more necessary than B. Degrees of belief are grades of modality. Having said that, there are questions on two sides. First, it is of course natural to think of a dual to necessity, a doxastic possibility operator. Second, we seem to miss rankings of non-beliefs. Even if a proposition is not believed, it may be more or less close to being believed. We shall distinguish non-beliefs in