Degrees of Knowledge
Belief comes in degrees—one can believe something more or less strongly. But knowledge does not come in degrees—one cannot know something more or less strongly. We have all sorts of words for degrees of belief (“certain”, “confident”, “convinced”, “of the opinion that”, “suspect”, “surmise”), but we have no such words indicating degrees of knowledge: you either know or you don’t know. In this respect “know” is like “refer”: we don’t have degrees of reference either. How then can knowledge be a type of belief—the true justified kind? If that were so, wouldn’t it have to be capable of degrees, just like belief? If knowledge were meritorious belief, it would have to possess degrees just like belief and merit, but we can’t say that someone strongly or weakly knows that p. Thus the traditional analysis has to be mistaken.
It might be replied that truth doesn’t come in degrees and that’s why knowledge doesn’t; the all-or-nothing character of knowledge reflects the all-or-nothing character of truth. But (a) that is not obviously true of truth, since we do sometimes talk of sentences (or propositions) being more or less true; and (b) the all-or-nothing character of truth doesn’t prevent belief from having degrees—so why should it have that effect on knowledge? If one can weakly believe a proposition that is definitely true, why can’t one weakly know a proposition that is definitely true? Nor can justification be the source of the difference between knowledge and belief, since it too admits of degrees. The all-or-nothing character of the concept of knowledge comes from the concept itself not from anything extraneous to it.
Yet meritorious belief appears to have something to do with knowledge; it is not irrelevant to the concept of knowledge. It is just that it can’t analyze knowledge: we can’t paraphrase a knowledge claim by using the concept of belief in the traditional way. But knowledge does somehow depend on the state of a person’s beliefs: what you know is a function of what you believe and how you believe it. Generally, if a person has a true justified belief that p, then he knows that p (putting aside Gettier cases). We thus seem to be heading for a paradox: knowledge can’t be a type of belief because it doesn’t come in degrees, but it must be a type of belief because that is what it depends upon. Here is a possible way out: knowledge supervenes on true justified belief, but it is not identical to true justified belief. The property of knowing is not identical to the property of having a true justified belief, but the former property depends (exclusively) on the latter property. We have dependence without reduction, as in other instances of supervenience. The traditional definition (sic) of knowledge confuses analysis with supervenience; the former does not follow from the latter. 
This diagnosis fits the account of knowledge I have suggested elsewhere, namely that knowledge is a relation to a fact and belief is a relation to a proposition.  The two states take different objects, which is why we can say, “John believes the proposition that p” but not “John knows the proposition that p”, and why we can say, “John knows the fact that p” but not “John believes the fact that p”. The state of knowledge refers to a fact but the state of belief refers to a proposition: the thing known is a fact but the thing believed is a proposition (the two states have different “intentional objects”). This theory fits the present diagnosis because it regards knowledge as conceptually separate from belief, involving a quite different relation (to a fact not a proposition), but it allows that there is a dependency between knowledge and belief. The state of being in the knowing relation to a fact supervenes on the state of being in the belief relation to a proposition (a true and justified belief), but the states are not identical or analytically equivalent. You get into the knowing relation to a fact by getting into the appropriate belief relation to a proposition (compare the mental and the physical), but we can’t reduce the former to the latter. Knowledge is thus not a species of belief, which is why it doesn’t come in degrees; but it does supervene on belief in the way articulated by the traditional definition. And intuitively, we have no wish to say that one can know a fact to one degree or another (just as we have no wish to say that one can refer to a thing to one degree or another), while we are more than happy to allow that propositions can be believed to this or that degree. Thus the factive theory of knowledge is consonant with the point that knowledge doesn’t come in degrees, while the doxastic theory of knowledge is inconsistent with that point. Still, there is a systematic dependence of knowledge on belief; it is just that the dependence is a matter of supervenience not analytic equivalence. 
 Supervenience specifies sufficient conditions not necessary conditions, so it is consistent with it to suppose that there can be knowledge without belief. This is desirable, since there are convincing examples of knowledge without belief (some animal knowledge, some unconscious human knowledge). A mongoose might know (the fact) that a snake is nearby without having any belief to that effect, belief being a matter of reasons and rational deliberation not instinct or direct perception.