Thank you for the criticism; let me see if I understand your challenge correctly.  O = observation, K = knowledge.

Eloise wrote:

I don't think this argument can work Quasar, because in the first few paragraphs you have stated that contiguous observations are sufficient evidence for contiguous observations and not sufficient evidence for knowledge.

O implies O

O does not imply K

Eloise wrote:

But here you have moved to say that a lack of observation constitutes a lack of knowledge,

not O implies not K

Eloise wrote:

this is contrary to your first statement from which it would follow that lack of observation is lack of observation, and nothing more is evident.

not O implies not O (derived from negation of O implies O, though also apparent through an identity axiom)

And you're claiming that these 2 statements are logically contradictory?

not O implies not K

not O implies not O

That said, I'll acknowledge that within the set of logical statements you're listing here, neither O nor not O implies K, which I would agree with using the strictest definition of knowledge.  I tried to indicate this in my previous post in the sense that regardless of what I observe, I can never know with certainty the true underlying nature of the thing being observed... all I can know is how it appears to me in my observation.

Eloise wrote:

To make observation and knowledge synonymous is to make an inductive statement that knowledge will always be observed by the knower. (NB you've already contradicted this statement in having said that knowledge must necessarily exist in the absence of contiguous observation.)

I think here you're referring to where I propose that, since knowledge in the strictest sense is unattainable through observation, that it's epistemologically justifiable to assign 100% confidence in the proposition that our observations reflect a reality which obeys some set of consistent rules. 

I want to be clear that I in no way propose that observation should be considered synonymous with knowledge in the strict sense that I used it in my previous post.

That said, if I understand you correctly, you're making the point that my line of reasoning makes the assumption that it's impossible to gain knowledge without making any observation. 

In other words, I understand you to mean that I don't properly address deductive reasoning like math and logic. 

But here is the problem with math and logic... the conclusions we draw from them only apply to reality to the extent that the axioms and givens used to derive those conclusions apply to reality.  So, you can select any axioms and given statements you want, and derive certain conclusions from them, and claim to have knowledge as a result... and I would agree that you have gained knowledge of a mental model which may or may not correlate to the real world.  In order for that to constitute knowledge of objective reality (which is what I was using the term knowledge to mean in my previous post), you have to already have sufficient knowledge of objective reality to know that reality is constrained by the axioms and given statements that your mental model is based on, and I challenge you to provide an explanation for how any perceived knowledge could be epistemologically determined to constitute knowledge of objective reality without the use of any observations.

I have absolutely no idea what you mean by the sentence in parentheses preceded by the nota bene.

Eloise wrote:

So to say, if we render a lack of observation insufficient grounds for knowledge then we have equated having knowledge with contiguous observation but you had earlier said  that observation only gives knowledge of itself thus is not sufficient proof of knowledge.

You're saying that not O does not imply K, therefore K = O? 

If so, I'm gonna have to call bullshit on that one.  To conclude that K = O, you would need to establish that O implies K (which by negation yields not K implies not O) and K implies O (which by negation yields not O implies not K).

However, in retrospect, I think I may not have been clear enough in my use of the term knowledge in my previous post, so I'll try to give a better explanation of what I mean in the hopes that it will clear up any misunderstandings we may be having.  I'm proposing that we know what our observations are, but that we can't know everything about the situations we observe.  I'm proposing that it's epistemologically justifiable to assume that, and to act as though, our observations are not random or part of an elaborate deception. 

I'm proposing that if our observations are consistent and useful that it's epistemologically justifiable to draw certain conclusions from them, but that it's also not epistemologically justifiable to draw certain other conclusions from them. 

I'm not attempting to provide a complete framework for which conclusions are justifiable and which are not, but to use my example of phase changes in water, I would say that it's justifiable to conclude that water will always transition through a liquid state in situations that are functionally equivalent to those situations in which water has been observed to behave that way, but that it's not justifiable to extrapolate that water will behave that way in all possible situations.  This avoids the problem of water skipping the liquid state under certain combinations of temperature and pressure.