@kennethamy,
Let's see if I understand this correctly. "Can I know that p, and p (actually) be false?" Here you are referring to the epistemic sense of knowing, i.e., in reality, if I know x, then x has to be true. So if I am reasoning inductively, and I am certain (objectively) to a high degree of probability that my conclusion is true, then am I correct to assume that I have knowledge? I assume you would agree even though it is certainly possible that a piece of evidence could turn up that would undermine what I believed to be knowledge. So in actuality, if I thought I knew based on the rules of good inductive arguments, and later it turned out to be false - it would not be knowledge.
If I am referring to inductive arguments, and I say that even though my conclusion is highly certain (objectively) it is still 'possible' that I am incorrect - this would still be using the word 'possible' in the epistemic sense, not the modal sense. The modal sense of 'possible' (logically or metaphysically) does not have to be reality - that is, it doesn't necessarily have to obtain - only in some possible world.
So when we use the word 'possible' in reference to what is epistemically possible, this is a 'mere possibility' as opposed to what is 'logically possible,' which is used in a modal sense. The logically possible, if true, is a necessary truth as opposed to a contingent truth - the former is contradictory if false, and the latter is not.
All triangles have three sides is a necessary truth. It is contradictory to say otherwise, thus it is not possible (in the modal sense or any sense) that triangles have more or less than three sides.
On the other hand, if I say that the moon is X number of miles from earth, this is a contingent proposition - denying it does not involve a contradiction, since it is not necessarily true.
There is a lot more to this, but is this the gist of what you are saying?