Statements are combinations of nouns and verbs and such like; Some statements are either true or false, and we can call these propositions. So, "The present king of France is bald" is a statement, but not a proposition.
Beliefs range over propositions. (arguably, they might be made to range over statements: Fred believes the present king of France is bald.)
Beliefs set out a relation of a particular sort between an agent and a proposition.
This relation is such that if the agent acts in some way then there is a belief and a desire that together are sufficient to explain the agent's action. Banno wants water; he believes he can pour a glass from the tap; so he goes to the tap to pour a glass of water.
The logical problem here, the philosophical interesting side issue, is that beliefs overdetermine our actions. There are other beliefs and desires that could explain my going to the tap.
We know some statement when at the least we believe it, it fits in with our other beliefs, and when it is true.
The "fits in with other beliefs" is a first approximation for a justification. Something stronger is needed, but material implication will not do.
Discard Gettier. The definition is not hard-and-fast.
It does not make sense to ask if we know X to be true; that's exactly the same as asking if we know X. The "we only know it if it is true" bit is only there because we can't know things that are false.
If you cannot provide a justification, that is, if you cannot provide other beliefs with which a given statement coheres, then you cannot be said to know it.
A belief that is not subject to doubt is a certainty.
Without a difference between belief and truth, we can't be wrong; if we can't be wrong, we can't fix our mistakes; without being able to fix our mistakes, we can't make things better.
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