Answer by Jim Ashby:
Once you know something is true, hope and belief are irrelevant.
Unless you already know it's false, then hoping something is true is merely a preference for a certain outcome. You're being objective: waiting for confirmation. Once you learn the outcome, it's assumed you'll accept it as new knowledge — whether or not your preference is confirmed.
On the other hand, hoping something is true when you already know it's false is known as delusion.
In contrast to hope, belief is subjective: if it were true it would be objective and not require belief. Sure, you can say you believe something known to be true is true, but that's, semantically inaccurate or confused (you're conflating the objective [truth] with the subjective [belief]). It would be more accurate to assert that you know it's true. It's important to use language as precisely as possible because you open the door to confusion by mixing meanings.
Because belief is subjective, others don't know why you believe what you believe unless you explain it to them. Knowledge, on the other hand, is objective and doesn't need explanation unless somebody is ignorant of that knowledge. Once you explain it, it's not up for debate: it's either understood or it's not.