Then I'm sorry, but your logic is flawed.  not O implies not K means that K must imply O (by negation) but makes no requirement about what O must or must not imply.  I could make the statement that O implies not K, and it would not be in contradiction with not O implies not K... although, doing so would mean that K must always be false.

I wouldn't say that knowledge gained from observation is not real... just that it's not complete, and that people often draw conclusions from observations which do not constitute knowledge.

Sort of... I don't mean to suggest that the behavior of reality is only consistent while it's being observed.  I do make the case that, since our observations don't reveal everything there is to know about the situations we observe, the consistency of our observations can be dependent on some variable(s) which we're unaware of and simply happen(s) to be the same each time we make an observation... and that, as a result of that, it's faulty to claim knowledge that the behaviors we observe will continue to be consistent in any situation.

Then again, your logic is flawed.  In order to show equivalence, you have to show that each variable implies the other.  Implication in only one direction is not sufficient to show equivalence.  Also, not O implies not O doesn't really demonstrate anything... it's just a usage of an identity axiom.  Although this:

seems to suggest that you think I'm using the flawed logic in question?  In which case, I can only assume that we have some confusion about what we're using K to mean.  This would be understandable, since I do suggest that, having made an observation, we know what we observed... however I tried to avoid using the term knowledge to refer to this in my first post to try to prevent such confusion.  When I referred to knowledge in my first post, and K in my second, I was referring to complete knowledge of the event being observed.  At no point do I try to equate observation to knowledge in that sense.

Fair enough.  Then, I'll state that I'm using knowledge to mean true, justified belief about reality.  A proposition which is imagined, but which is not supported by any observation whatsoever, may be true by coincidence, but can't really be said to be justified, since there doesn't appear to be any way (again, without observation) to distinguish it from other propositions which are imagined, not supported by any observation whatsoever, and are false by coincidence.

That is the point I'm trying to make, yes.  The logical conclusion that we get from extending this line of reasoning is that, no matter how many, and how varied, our observations are, we can never know that we've really understood the true underlying nature of the things being observed... and that we can never know that there's not some condition or variable that, if it were different, would cause our observations to be different.

The point where we disagree seems to be that I think that the observations do apply to an objective reality... just that they do not reveal the complete truth about that reality.  Even in the case of The Matrix or the brain in a vat, those are still objective realities which we would be observing, it's just that those objective realities would be very different from what we would expect.  It doesn't mean that our observations don't tell us anything about those objective realities, it just means that we don't know the whole story.

I think it would be useful to extend my water example here.  The person in the past, who has only observed state changes of water that pass through a liquid state, has observed water changing state from a solid to a liquid to a gas.  While he doesn't know that water always passes through a liquid state when transitioning from a solid to a gas, he does know that water is capable of being in any of those three states, and that it's capable of changing state.  He may not know why, how, or when it happens, but he knows that it can happen in the right circumstances.  With further observations, he might learn that temperature is a determining factor (because he can repeatedly cause the change by adjusting the temperature), but that's not enough to determine that temperature is the only factor.  He would be unlikely to discover that pressure is also a factor because, for the purposes of this example, he will die before fluid pressure is discovered.  Similarly, today, we know of two factors... temperature and pressure... but we don't know that there isn't a third, undiscovered, factor that we haven't learned to modify yet, and we don't know what temperature and pressure really are although we do know enough to be able to measure them and manipulate them under the circumstances we're familiar with.

We appear to be in agreement here as well.  That's why, in my first post, I tried not to use the term knowledge to refer to conclusions drawn from observations, but instead to say that we can determine epistemologically justifiable levels of confidence through observation.

I think you're using the term probability to mean the same thing, though I hesitate to use that term in this way, because to me probability represents a branch of mathematics that deals with hypothetical situations, was invented to determine useful odds for gambling, and appears to be regularly misunderstood and misapplied to try to justify things which it doesn't justify.  Specifically, probability is used to justify the belief that a single event can have more than one possible outcome, and that the likelihood of those different outcomes is accurately represented by a probabilistic model... and probability in no way provides any such justification.