"You knew I wanted to cash this cheque, so when we were walking down the highstreet why didn't you tell me the bank was round the corner? You knew that it was there."
"You knew I wanted to cash this cheque, so when we were walking down the highstreet why didn't you tell me the bank was round the corner? You believed that it was there."
Notice that the second example sounds a bit odd but if it means anything it is an accusation of deliberate deception. If you believed something it was there before you, in your mind.
If you know something, it is quite possible that you have never thought about it.
I'm now thinking the highlighted bit is quite wrong. I didn't like it when I wrote it. [But the example still speaks for itself.]
What would I say instead, though. If you believe a proposition, it is like saying you favour it as an instruction. If you believe there is a table in front of you then you act as if there is a table in front of you.
Compare:
"Yes, I know that I need to stamp the letter before putting it in the envelope."
With:
"Yes, I believe that I need to stamp the letter before putting it in the envelope."
Or worse:
"Yes, I believe with certainty that I need to stamp the letter before putting it in the envelope."
The first of these is a statement about the way the world is in relation to my place in it. It is not about the proposition, "I need to stamp the letter before putting it in the envelope." Suppose the letter is already stamped, I would not follow the instruction, because I have understood the purpose of the instruction.
The second example sounds a little deranged. It is like saying that I will stamp the letter before putting it in the envelope, regardlessly. It makes no claim to understand the relation I have with the world - only the relation I have with the proposition.
The third example is not closer to the first, but further from it. It sounds like, come hell or high water, I will stamp the letter before putting it in the envelope.
Knowledge is not about propositions - insofar as it is connected with propositions at all it is about the content of propositions - facts and states of play.