Knowledge, belief, and Anton's Syndrome
Mostly, when we don't believe something, we don't know it either. But arguably not always. The timid student thinks she's merely guessing, while in fact she knows. She knows, but she lacks the confidence required for belief. It would be nice to have an analysis of knowledge that allowed for such cases, but also explained why they are rare.
Lewis's analysis tries to do that. On Lewis's account, you know p iff your evidence rules out any relevant situation where ~p. Among the rules for what counts as 'relevant', the 'rule of belief' tells us that any possibility with non-negligible subjective probability counts as relevant. Now suppose you don't believe p. Then you give non-negligible probability to ~p situations. So you know p only if your evidence rules out all those ~p situations. Moreover, your present evidence 'rules out' a situation iff you have different evidence in that situation than you actually have. So if you have knowledge without belief, you must assign positive probability to situations where you have different evidence than you actually have. On a suitable understanding of evidence, those cases will be rare, because we are normally confident that we have the evidence that we have.
However, I think the relation between knowledge and belief should be even tighter than it is on Lewis's account. Yesterday, Fiona talked about Anton's Syndrome here at the RSSS, and it occurred to me that this is a particularly striking example. Patients with Anton's Syndrome are blind, but insist that they can see. Let's suppose (what isn't obviously true, but certainly possible) that these patients don't have any sensory experiences, but are really convinced that they can see. Then on Lewis's account, they would count as knowing that they cannot see, for their evidence is incompatible with any possibility in which they can see. But that seems wrong. It is very natural to say that people with Anton's Syndrome do not know that they can't see.