A different objection, following Eliezer's PS, is that:

Between me and a red box, there is a wall with a hole. I see the red box through the hole, and therefore know that the box is red. I reason, however, that I might have instead chosen to sit somewhere else, and I would not have been able to see the red box through the hole, and would not believe that the box is red.

Or more formally: If I know P, then I know (P or Q) for all Q, but:

P => Believes (P)

does not imply

(P v Q) => Believes (P v Q)