If we try to use premises to arrive at a conclusion, how can we know that we have arrived at the primary premises? For instance the idea of self-ownership. It is taken as a self evident concept by most. If our premise is based on God given rights or natural rights, would not we be employing a premise which cannot be proven?

You don't arrive at the premise. You arrive at the conclusion through the premise. You create each premise from either evidence, assumption, deduction, or axiom. But the process of logic is to then take your premises and lead them to a conclusion that is at least valid if not sound. The major premise of any argument or logical statement is the premise you intend to prove. It is usually the first one stated. The minor premises should support the major premise or counter any outcomes that might negate your major premise.

For instance:

All cats are four-legged (major premise)
Garfield is four-legged (minor premise)
Garfield is a cat (valid conclusion)

This is not a sound conclusion because Garfield could be a dog. There is no way to tell the fact with the premises provided. First you need to understand that a premise is what you seek to prove. You create the premise, so it is only as faulty as you or anyone makes it. Premises do not self-create.

You know what the primary (major) premise is because you created it to be your primary premise. If your major premise is in doubt, then you probably shouldn't have it as your major premise to begin with. The system of paraconsistent logic and the modern logic principle of dialetheism explains how two contradicting premises or arguments can both be true at the same time.

"All cats have four legs is false."

Not exactly. This is what is referred to in logic as a valid statement. Valid statements need only be analytically agreeable to be true. That means that it has to satisfy the need of the immediate argument which is isolated to the statements (premises and conclusion) made. It is not a sound statement, however, because it is not in synthetic agreement with reality. Soundness requires that a statement be realistically true.

All in all, if you conjure up a set of premises, it is up to you to find the logic that links the various points of information that leads to a sound conclusion. If you fail to do so, the premises are either unsound or you are not up to the task. There is no system of logic whereby you just input any premises you want and it spits out a sound result for you. Logic requires applied intelligence. It you had a set of predetermined premises, you could use computational logic to produced a series of predetermined outcomes or conclusions, but that is a close as you get to an automatic answer in logic.

In the technical sense of terminology a ‘premise’ may be different than a ‘conclusion,’ but in the practical sense a premise is a conclusion and a conclusion a premise. For instance, if you set up a two premises which give you a conclusion, that conclusion can then become the premise for another conclusion. At some point we deem the premise an axiom. But how can we know that the axiom is not just a conclusion with another premise behind it?

An axiom / postulate is a premise or starting point of reasoning. On the above context, the use of the word "axiom" is not strictly correct. It is axiomatic only in that you judge it to be so in a particular argument. But this is not a true axiom. It's a principle or fact. We sometimes just used axiom as a synonym for fact, but that doesn't change the original definition of an axiom. As I stated earlier, axioms by nature are not ambivalent. If you can question their soundness, they are not axioms.

For instance, as an extension to the axiom of choice, given two sums, one sum will always be equal to or greater than the other sum. Try as hard as you like, you will have an impossible time disputing this without severely warping the physical laws involved. This is a true axiom. Axioms are defined as being self-evident and indisputable. If you can reasonably question an axiom, then it is not a true axiom. If it is a true axiom, then it shouldn't be possible to be ambivalent about it.

It is possible to link multiple premises and conclusions together in chains, however, a statement can only serve as a premise or a conclusion for any singular argument. It can't be both at the same time for any particular argument.

If you find yourself questioning the reality and natural law behind the axioms, questioning whether up is truly up and down and truly down, then logic is not your thing. Logic requires a certain level of acceptance of external reality. In other words, you have to be a realist in order to be good at logic.

The principles of logic have worked very well for us for about 2500 years. People will stop using to its methods once it stops working for us. That hasn't happened yet. Not to a level that would invalidate the process to any significant degree. I'm not the first to ask these questions. Many a logician has gone over them a thousand times previous.

 You don't need to define reality. That's an unnecessary tangent. If you don't know what the scientific / logical definition for reality is, I repeat, if you find the need to redefine or deny reality all the time, logic isn't for you. An idealist trying to understand logic theory is like a vegan trying to find something to eat on Burger King's menu.

A principle or fact can change with the conditions surrounding it. So facts are not axioms by default, and a premise can be based on either. It's a fact that the sky is blue. It is also a fact that the sky is colorless. Both are true, but both are dependent upon different conditions. It's a fact if that information can be shared, and the individual you share it with is able to independently verify said information. Facts are not required to be objectively absolute.

Again, something being an axiom and something being deemed axiomatic are two different things. It's like calling something absolute to demonstrate just how important you think it is. Like love is an absolute truth. Is it really?