There is no “really” knowing something. There’s just knowing. There isn’t any underlying truth we’ve got to uncover; the moment we find it, we know it. Sure, we were wrong before, but now we know better.

Seriously, though. There is nothing gained by intensifying the verb “to know”. It isn’t as if people ask, “Do you REALLY know that?” and the response is, “Oh, I was just talking about everyday knowledge.” Knowledge is knowledge: the ability to demonstrate belief. We know because of two things:
- We agree on a method of demonstration; (for the philosophically inclined: the pull towards foundationalism)
- We agree on a common core of understanding. (for the philosophically inclined: the pull towards coherentism)
Knowledge is then revealed by the demonstration of what is said to be known referencing existing understanding (or in the case of more complicated assessments, existing understanding and previously revealed knowledge). More can be known than propositions about the world; tasks can also be known, and there are ways to demonstrate one knows them (usually by performing the task). Knowledge is not strictly a matter of logic—though that might be one way we demonstrate it. Knowledge is not strictly the consequence of demonstration—for we can demonstrate a grasp of the tools used to demonstrate in various contexts (think on the difference of “knowing a theorem” of arithmetic by demonstrating it through it versus what you’d show someone to demonstrate you “know arithmetic” and the distinction I draw might become clearer).
We cannot come to understand the activities of perception until we are taught the word “perception” and the various ways humans look at it already… by then it is too late[sup]†[/sup]. At that point the only thing that dispels one type of demonstration is a “better” demonstration, but we are never without knowledge. The best that can be said is that we attempt to disregard our naive tendencies to assume what we already know and regard the world through eyes that don’t see objects and colors but simply raw sensation. The utility of this method has been thought on by phenomenologists, most famously Husserl, Heidegger, Merleau-Ponty, and Sartre. The conclusions they came to from this method vastly differed from Descartes who purportedly undertook the same, or a similar, method (even though Husserl was in many ways inspired by Descartes). What this indicates—I don’t feel like speculating on here (my opinions on Descartes can be found in at least two threads here in GD).
Importantly, there are some things that we might come to say we know but that are not strictly knowledge in the sense I have described above. These are the sorts of things people often reference when they intend to intensify the verb “to know”. For instance, how does one really know other people exist, or how does one really know whether this is a hand (that is, my hand). These sorts of brain-smashers are best handled with the above demarcation of knowledge. We cannot say we know some things because of criteria (2) of knowledge: the common core of understanding. The elements of (2) cannot be known; they are used to show what we know (I do not demonstrate that I have a hand when I draw; I demonstrate that I know how to draw with my hand).
†[sub]If you ever see me irritated with people who insist on only knowing things through perception, this is why. I find the position to be completely untenable. Hume’s exhaustive view of the primacy of perception remains, I feel, completely unchallenged. Either his path was the proper path, or his assumptions were wrong. I believe the latter, and require more of knowledge than sensation, which is really quite pitiful in its powers. For the artistically inclined, impressionism offers, I think, a fine visual aid for understanding why sensation alone (in this case, vision), is woefully inadequte to account for any knowledge whatsoever.[/sub]