If it's true that "I know it's raining", is it necessarily also true that "I know that I know it's raining" and "I know that I know that I know..."?
P.S. A similar example is "I try to try to try to solve this problem". Maybe these cases are of no significance, but I find them interesting. Is there a category for such examples in semantics?
It's a bit unclear from your question, but I think you are highlighting "recursions". Recursions in general aren't that philosophically interesting past the second instance due to the following:
(1) is just our normal picture of knowledge talking about facts in the world.
(2) is interesting philosophically because it looks at a type of reflective knowledge where we can have knowledge of our knowledge. This is important for the reasons Geoffrey Thomas's answer highlights. Specifically, it's not the same as just knowing that thing. Moreover, it demonstrates a type of self-awareness that reveals something interesting.
(3), however, doesn't tell us anything new. Because it is exactly like (2) as a form of knowledge -- knowledge that you know something. The third degree doesn't change the type of thing it is, so it's just a trivially different version of (2). And generally, there's not much to be gained by going to the third level here.
There's two reasons for that. First, I know the the things that I know I know. What does knowing the things that I know I know add to this? Nothing but complexity really. Second, this pushes us towards a bad infinite regress -- which unproductively multiplies entities because we already know everything in it and know that we know everything in it.
A similar picture arises with will. Some theories of autonomy differentiate between a first and second order will (See this article by Holton on Frankfurt). On such theories, first order volition is the things I want immediately and second-order volition is something I use to regulate the first-order volitions. You can add third-order volition but it's not necessary because that's really just another species of second order volition (i.e. volitions about my volitions).
Case 1
A knows that p
A knows that q
It does not follow that A knows that (p and q) since A may never have considered these two items together.
Case 2
If we assume that A knows that p iff :
1 A accepts p
2 A has adequate evidence for p
3 p is true
then it may be that case that p is true, A accepts that p and has adequate evidence for p but does not recognise the adequacy of his evidence. In this case we can say that A knows that p because conditions 1 - 3 are met. Yet A does not know that he knows that p.
You're invoking the closure principle, which has various problems that leave it untenable in many cases. It does not follow that if you know A that you know that you know A. For instance, I saw my keys on the counter earlier but now that I'm looking for them, I can't remember where they are; later, however, I remembered. On the multiple choice question, I knew the answer was C, and it turned out I was (right/wrong). There are many things to be said of such examples, but for whatever problems there may be there, this is a well discussed issue in Philosophy and should be researched. Generally, what we mean by "know" is what makes any difference that makes a difference, and then in most cases is trivial.
