There has been a lively debate in recent years about the relationship between graded belief and ungraded belief. The debate presupposes something we should regard with suspicion: that there is such a thing as ungraded belief.
Compare earthquakes. I'm not an expert on earthquakes, but I know that they vary in strength. How exactly to measure an earthquake's strength is to some extent a matter of convention: we could have used a non-logarithmic scale; we could have counted duration as an aspect of strength, and so on. So when we say that an earthquake has magnitude 6.4, we characterize a central aspect of an earthquake's strength by locating it on a conventional scale.
Now, besides the graded term (or concept) 'earthquake with magnitude x', we also have the ungraded term (or concept) 'earthquake'. Should we conclude that there are two kinds of earthquakes: graded earthquakes and ungraded earthquakes? Clearly not. It would be silly to investigate the relationship between graded and ungraded earthquakes, or to wonder how our concept of an ungraded earthquake relates to our concept of a graded earthquake. We don't have a concept of an ungraded earthquake, in any interesting sense. We don't have a concept of an earthquake that has no degree of strength.
What one can reasonably study is our usage of the ungraded term 'earthquake': what kinds of things are properly called 'earthquakes' in a given conversational context? Is that just a function of the tremor's magnitude, or is it also sensitive to duration, to the damage caused, etc.? These are reasonable questions, but they are really questions for linguists, not geologists.
Beliefs are a lot like earthquakes. Like earthquakes, beliefs plainly vary in strength. I believe that I am employed by the University of Edinburgh and that David Cameron will resign in October, but the first belief is stronger than the second. Whenever someone expresses a belief, it makes sense to ask about the strength of that belief. It would be bizarre to respond that the belief in question is an all-or-nothing belief and thus does not have a degree of strength.
Like in the case of earthquakes, there are somewhat different criteria for measuring the strength of a belief, and there's obvious conventionality in the choice of a scale. Decision theory states how different degrees of belief manifest themselves in rational choice, suggesting that degrees of belief could be measured by their effect on behaviour. From the other direction, Bayesian epistemology and confirmation theory state how rational degrees of belief reflect an agent's evidence and how they should change over time, suggesting that one might measure an agent's degrees of belief by looking at their evidence. It is not obvious that these two approaches perfectly coincide, and how each of them relates to our intuitive notion of strength of belief (or degree of confidence). Perhaps there are several, highly correlated, dimensions of belief strength. But that doesn't change the fact that beliefs vary in strength.
So we should be suspicious of talk about ungraded beliefs. We have an ungraded word 'belief', and it's worth asking how that works, but it's not clear why that should be any more relevant to epistemology than the semantics of 'earthquake' is to geology.
To be fair, when people discuss the connection between full belief and degree of belief, they often outline a job description for full belief. The job description varies from author to author. Some suggest that having all-or-nothing beliefs (and desires) in addition to graded beliefs (and desires) can be useful for agents with bounded resources because computing probabilities and expectations is hard. It's true that computing probabilities and expectations is hard, and it's a good question how the ideal of Bayesian rationality has to be compromised in order to take into account computational limitations. There is nice work on this in computer science. It is certainly not obvious that the best alternative involves ungraded beliefs and desires.
There is another job description for full belief that I like a lot better. It has been outlined by Quine and Carnap and eloquently defended in Patrick Maher's Betting on Theories. My own take is a little different from Maher's, however.
The basic idea is to define full belief in terms of conditions on linguistic assertion. Kripke wrote (in "A puzzle about belief"):
(KP) If a normal English speaker, on reflection, sincerely assents to 'p', then he believes that p.
Kripke's principle links an ungraded notion of belief to linguistic assertion. The principle only states a necessary condition on full belief, but we may strengthen it to a necessary and sufficient condition. Roughly like so:
(KP+) An agent believes that p iff she is disposed to sincerely assent to a translation of 'p' into a language in which she is competent.
(KP+) is obviously inadequate as a general analysis of the English term 'belief', but I think it approximately captures one core usage of that term.
Other terms from classical epistemology can be understood along similar lines.
"Knowledge is the norm of assertion". -- Let's take that to define an ungraded notion of knowledge. Very roughly:
An agent knows that p iff she would conform to her linguistic norms by asserting (or assenting to) a translation of 'p' into her language.
The idea is that 'knowledge that p' (on this usage) is a label for whatever the linguistic norms require for an assertion of 'p'.
It is clear that the norms of assertion can be divided into several parts. If someone has strong but misleading evidence that it is raining and falsely asserts "it is raining", they are liable to criticism, but their fault is quite different than if they had uttered the sentence without having any evidence that it is raining. In the latter case we can further distinguish between unwarranted but sincere and insincere assertions. Again it's useful to have labels for these components. Let's understand 'truth', 'belief', and 'justification' as such labels. Thus (very roughly) a "belief that p" is "justified" if the agent's internal state conforms to speaker's linguistic norms on assertions of 'p'.
When I read classical (non-Bayesian) epistemology, these definitions often help me to follow the discussion. On that interpretation, classical epistemology is really part of linguistic pragmatics.
The interpretation also provides an answer to the above suspicion that there is no ungraded concept of belief. (KP+) defines such a concept. And it makes good sense to ask how the pragmatic notion of belief as defined by (KP+) relates to degrees of belief. More generally, we can ask about the constraints which the norms of assertion put on an agent's degrees of belief. It seems unlikely that "justified belief" is simply a matter of what rational degrees of belief the agent should have: the norms of assertion are plausibly sensitive to conversational background, to what's at stake, to whether errors can be excused by unusual circumstances (as my colleague Martin Smith has argued), and so on. Moreover, "justified belief" is a hyperintensional matter, while degrees of belief are merely intensional. So the connection between (the pragmatic notion of) justified belief and (the epistemic notion of) rational degree of belief is bound to be complicated.