I think we're having a semantics issue.
- Philosophim
 
These are not semantic issues they're the meat of these questions. We have completely opposing world views. 
 
The world has order. This order is the predictive relationships of environments.
Environments are unstable. Organisms that evolve simple genetically coded behavior are less likely to survive that those organism that evolve Associative Learning Mechanisms (ALM) to gather  information of their environment.
 
Species process the ALM of the predictive relationships in their environment by evolving Associative Application of Information(AAI) mechanisms. 
AAI is rationality
information A with information B results in -C. C, or +C.
 
This is the rationale; genetically encoded in life long before there were any humans to use it to create logical systems.
 
Rational thinking requires logic. - Philosophim
 
No, logic is a creation of rational thinking.
In the example of the pre-verbal toddler he has neither language or belief. His activity is a genetically encoded AAI mechanism.
Encountering information, associating information and testing the association.
 
The example of the toddler is not an philosophical abstraction created to illustrate a point. It's a real world phenomena to which I am giving explanation and I don't see that process in your paper. There are no real world testable conclusions in your paper.
 
But one problem in epistemology is determining the validity of different types of irrational thinking.
- Philosophim
 
I disagree, real irrational thinking is dysfunction, brain damage or chemical imbalance.
Other than that, thinking follows the rational AAI pattern.
Criticism of validity is at the points of information, association or testing.
 
A foundation for epistemology needs to produce testable claims about the phylogenic, ontogenic and cultural environments.
 
Such a claim must stand under criticism: 
On its utility in problem solving. (Does it work?) 
On its internal coherence.  (Is it self-contradictory?)  
On its external consistency.  (Does it 'fit' in a framework of other claims about the world?) 
On its semenality.  (Does it/can it lead to new/more precise claims?)
 
I find no utility in the system you propose. Your answers to criticisms are self-referential, externally inconsistent or Argumentum ad dictionarium. Being devoid of testable claims it has no semenality.
 
The paper gives us no information promoting an understanding of the general nature of knowledge or the species-specific human nature of knowledge creation and use. It's answers to these question that an epistemology can be founded on.

Until you produce a testable conclusion on the nature of knowledge, not the nature of a logical system, you have no foundation for an epistemology.

"What am I testing if I have no belief?
After combining the first two bits of information, the child had a hunch…."
- Philosophim
 
The information is being tested.
The "hunch" is the instinct to learn born of evolutionary heritage made rational by  evolving in a rational world.
 
 "…alternative to logical thinking, is illogical thinking." - Philosophim
 
Logical systems are human creations based on the instinctual rationality formed by evolving in a rationale world. Rational thinking does not require a formal logical. The majority of rational thought is simply utilitarian, knowing something worked before and believing it will work again, from this logical systems are born.
 
This is a creation of a formal theory of knowledge that can be used to explain why that child knows in terms that can be reapplied to any case. - Philosophim
 
Then you should be able unpack your conclusions from the system and return to the field where you encountered a sheep and apply them to the sheep (or any animal you're familiar with) to explain its knowledge system. (Non-human learning and knowledge)
 
How would your conclusions explain people holding and acting on beliefs that they admit have no actuality to support them; in some case admitting that those beliefs are disproved? (Belief as emotional attachment to a social group.) How does it apply to Scientific Methodology's creating knowledge claims as a social activity?
 
"While you may have trouble with the premises, you seem to not be arriving at my conclusions." - Philosophim
 
Well yeah, of course. You've carefully designed an enclosed abstract logical system where the design insures the premises support the conclusions. The only lines of criticism open are the premises.
 
Your conclusions, whether they have real world validity or not, are not relevant to my criticism, that an abstract formal logical system, no matter how carefully constructed, cannot be imposed on the world as foundation for an epistemology.
 
It should be obvious by now that my strict-materialist evolutionary naturalism will find few points of agreement with logical idealism.
If you are only interested in someone to parse the logical formulae and not interested in systemic criticism feel free to say so. 

I have to agree with Jarmo here. It's the validity of the premises I question.

"That sentence is not a conclusion, only a first premise I consider." - Philosophim
And
"Knowledge occurs after you run a belief through a process." - Philosophim

I will unpack the example of the toddler and the water bottle more fully.

He feels a wet spot on the floor - information.
He drinks from the bottle, it's wet - information.
He shake a few drops onto the floor and feels the result - information testing
.
Tested information is knowledge. The knowledge claim is knowledge plus belief in the validity of the claim. Starting with belief ignores the fact that a belief can be held without any validity. Knowledge proceeds belief.

I know you try to work through this farther on in the paper. Consider the meta-criticism I made.

You are attempting to create an epistemology based on a formal logic system. A formal logic system cannot serve as a foundation for an epistemology.

A formal logic system is itself a knowledge claim.
A formal logic system can validate logically proofs that have no actuality.
A formal logic system creates logical proofs that only prove the system and only within that system.

Logic is one of the best tools we've created to refine information into ever more precise knowledge-claims. Like all knowledge-claims it has limitations, particularly when comes to creating knowledge-claims.

I have problems beginning with the first sentence.

"Any discussion of knowledge must begin with beliefs.  A belief is a will, or a sureness reality exists in a particular state". 

This claim seems to be that you were born with a set of beliefs that you that you've tested to create knowledge.

'A thirteen-month old child is playing by himself on a rug. He encounters a wet-spot on the rug next to his bottle. He picks up his bottle and sucks on it. Then he rubs the wet-spot again. He shakes his bottle until several drop fall out onto the rug. He rubs his hand over the new wet-spot.' A Young Child is… (A documentary film by Barry Hampe)

In his book ‘Making Documentary Film and Reality Videos’ Hampe writes about obtaining this footage. They wanted a piece on walking and talking in toddlers. The crew simply focused their cameras on this young boy and waited for him to walk or talk. It wasn’t until they began to edit the film that they found this piece of natural learning. The child perceived a wet-spot in his environment and it aroused his curiosity. Experience leads him to associate wet-spot with bottle contents. He tested this idea and confirmed its’ correctness.

This is how knowledge-claims come into being. We wonder, we test; we store the information away for future use and testing. knowledge-claims are not created by belief-claims. Belief-claims are created as justification for knowledge-claims.

Belief-claims can be tied to objective proofs (Certainty), subjective proofs (Certitude) or no proofs at all. It's knowledge-claims that require an objective justification to be valid.

The main problem is logic as Justified True Belief (JTB).
Logic is a knowledge-claim in itself. It is an enclosed system with specific rules. Correctly follow the rules will create a logically valid conclusion, but its validity is only within that enclosed system. Logical proofs only prove logical systems, ie, the problem of self-reference.
Then there's the Gettier Problem to answer. Wikipedia has a primer on that.

HOW SOMETHING COMES FROM NOTHING
An unscientific creation myth of a scientific concept. Guaranteed to contain as little science as possible and be a more philosophical view of existence. A metaphoric for an unscientific understanding of a scientific concept, for those of us who learn science from watching the SYFY channel.

How can something come from nothing? Everyone has heard this question.
To understand how something comes from nothing you must first understand what
you mean by 'nothing'.

If you ask most people, "What is nothing?", you basically get a reply that nothing is nothing, just nothing at all.

They cannot describe any qualities of Nothing because what they are trying to describe is an abstract philosophical concept. Abstract philosophical concepts exist only as imaginary qualities and you cannot imagine the quality of no-qualities.

To understand Nothing you have to reject the abstract philosophical concept and look at Nothing 'in the wild' where it has actually existence. Those who do this have found some interesting things.

Nothing, 'in the wild', is a state existing without energy or mass.

This state has the quality of being unstable. In fact all states have the quality of being unstable.

This quality of being unstable has the effect of collapsing into different unstable states some of these states achieve temporary qualities of equilibrium.

An effect of some of these collapses is zero-point-energy and virtual-particles. When enough of this occurs in proximity it generates, an effect of mass, causing a force of mutual attraction a kind of pseudo-gravity. This mutual attraction holds and attracts more to it. A new state of equilibrium grows, causing heat and a new unstable state that grows until there is a sudden violent, hot collapse causing another new equilibrium.

Nothing hasn't disappeared in this new state it has merely separated into localized points of positive and negative. Points of localized mass, matter and anti-matter and positive and negative energy. If you subtract the - from the + the result is 0; everything is made of Nothing.

Any question of some existing dichotomy between Something and Nothing falls with the concept that something is nothing, nothing separated into positive and negative parts. No creation of energy or mass has occurred there is just a state of separated nothing. There is no 'Why' here only the effects of unstable states.

Objects are effects of this new state. Objects have mass and mass causes gravity. Three dimensions exist, dimensions being directions of movement for objects. Objects moving in 3-D is what time is.

We are lucky in that this particular state has the qualities that support the existence of organic life forms and the kind of unstable environment that allows evolution. We can only imagine the billions or trillions collapses that may have come and gone before us.

Thanks

So another JTB problem. (see also; The Gettier problem)

All justification is open to criticism.  All justifications pursued far enough lies on an unjustified/self-referential statement.  I don't follow the Popperian into a denial of Justification.  (Karl Popper, W.W. Bartley et. al.).  I do deny that justification equals certainty.  Just as Natural Selection doesn't evolve the optimal solution but rather one that is satisficing under current conditions; knowledge claims don't rest on absolute certainty but on a satisficing claim under current conditions.

So how do we know a satisficing claim?  The answer is:
(1) how well it stands under criticism. 
(2) On its utility in problem solving. (Does it work?) 
(3) On its internal coherence.  (Is it self-contradictory?)  
(4) On its external consistency.  (Does it 'fit' in a framework of other claims about the world?) 
(5) On its semenality.  (Does it lead to new/more precise claims?) 

From this we don't create some truth or certainity, but rather a judgment tool. We are no longer looking for some absolute fixed point, some Wittgenstein 'hinge' on which everything turns.  We are minimizing error, not creating certainty.

As for the problem of the probability of your existence, since you and I are compelled to act as if you are an actuality I can give no validity to the idea that you are not an actuality.
Probability of non-existence 0, probability of existence 1.