There is an ongoing debate about how to connect categorical beliefs to graded beliefs. That there must be such a connection has seemed a plausible premise, given that categorical and graded beliefs live in the same space, colloquially known as ‘a human's head’. Also, given that degrees of belief are degrees of belief and categorical belief is belief, how could there fail to be a connection? While these observations do not imply that the connection must be definable in logico-mathematical terms, that it is thus definable has been deemed a further reasonable assumption by most formally inclined philosophers.
The pre-theoretically most plausible formalization—to conceive of categorical belief in a proposition A as A being believed to a ‘high enough’ degree—has long been known to run into paradoxes, specifically the Lottery and Preface Paradoxes. At least that is so given some pre-theoretically equally plausible ideas about categorical belief, most notably...
