Lets begin by clarifying some ideas about freedom and capability.
I define "will" in the classical definition of "a power of the soul" that is specifically the power to make choices. In classical platonic and Christian philosophy the object of the will is love. The object meaning it's purpose or the thing it is meant to do and pursue.
I think that Intellect is also important to this discussion. I define intellect, again in the classic definition, as the power of the soul which knows things. This includes not just abstract ideas, but knows real things, both physical and spiritual as they are in themselves. In classical platonic and Christian philosophy the object of the intellect is Truth.
When I speak of "freedom" I don't mean an utter and complete lack of constraint or limitation. I mean free within the limits of your nature.
Using the example of a person and our ability to move. We have the power of locomotion. We can move ourselves around. However, there are limits on that power. I cannot fly, for example. The fact that there are limits of this kind does not mean that I am not free to move. My nature limits the ways in which I can move, and so long as I can move in those ways, I am free because I am free to be what I was created to be. I'm free to be what I am. To believe that I would need to be able to fly, to move freely would be something like thinking that I need to be a bird, in order to be a free human. This is actually antithetical to real freedom, because if I were to become a bird, I would no longer be human, and no longer be fully myself. This would actually be a huge restriction and loss of myself. This is actually a mistake that is growing more and more common in our world today.
However, though human beings are free by nature to move according to their nature, it is entirely possible that I can become less free by further constraining my ability due to accident or poor choices. For example, I could become injured, which might impair my ability to use my natural freedom. Or I could begin to lose capability through lack of exercise. In these cases my natural freedom is impaired, but not entirely lost.
It is also worthy of note that I can fail to reach the full potential of my freedom because I can move better if I practice at it and exercise etc. My full potential can fail to be realized, just as I can degrade my ability from it's starting point.
Lastly, it is possible that I can lose my freedom either by becoming so broken physically that I can no longer use it, or by being externally restrained, such as being put in prison, or in a straight jacket.
The principles here are as follows
1. Freedom is not unlimited, but exists according to our nature. True freedom is the freedom to be fully yourself, to the best of your potential.
2. We can limit our own freedom either by not striving to reach our full potential, or by actively reducing our ability through bad choices. It can also be reduced by circumstances that happen to us.
3. We can lose our freedom by becoming so broken that we no longer have any capacity to use it, or by being externally constrained.
For random, I'm just going by the dictionary definition:
the definition of random
"proceeding, made, or occurring without definite aim, reason, or pattern"
Specifically "without reason".
For each atomic, indivisible event E in the decision making process, either E is determined, or E is not determined. So "free will" and "random" have to fit together in the same category.
Bob is free from compulsion to force my brain to act in a loving manner. He loves God, he loves others, so he wants to make my brain spend this Saturday volunteering at the soup kitchen. Is he truly therefore capable of choosing otherwise come Saturday morning then? If so, what caused him to do so? This is where I struggle understanding libertarian free will.
The reason that I balk at the use of the term random is that to me random conotes the lack of intellect. Which would more or less be similar to saying "without reason" in the sense of reason as thought. It might not be the same as saying "without reason" if by "reason" you don't mean thought, but instead mean cause, or explanation.
In the material world, in classical physical mechanics, randomness is impossible. There is only the appearance of randomness because the system is so complex that we don't understand it and there are so many unknowns that we can't predict.
In the realm of mind, it is debatable whether randomness is possible or not. It is likely not provable either way because there is too much of a black box that we can't see beyond. For example, no one really knows where ideas come from. Some ideas, of course, are prompted by the world around or by other people etc. However some apparently come out of no where. At least no where that we can see.
You could call this random, but we really don't know what it is. If you believe the traditional Christian spiritual worldview, then it is likely that some of these ideas are inspiration from both angelic and demonic forces. Perhaps the rest are inspiration directly from God? We simply don't know.
This deals with ideas, not will directly.
The will, however, is intimately connected and intertwined with the intellect. They are distinct powers but they work together very closely and they direct each other.
When we make choices, our will almost always makes reference to what we know (ie the intellect). However, the intellect is also directed by the will because the will chooses what ideas and information we accept and what we reject.
As a historical note, there is a theological school of thought that places the divine will absolutely prior to and superior to the divine intellect. In other words, God does not will according to what he knows, rather he knows according to what he wills. In this view God's will is essentially arbitrary and what you would call random. This school of thought is known as Voluntarism. It was a response by philosophers who thought that classical views of intellect and will placed too much restraint on the freedom and sovereignty of God. They thought that if God wills according to what he knows, this implies that God is bound by knowledge, which then must be of something higher than God.
The classical view is that God is constrained only be his own nature. This is to say that the only thing God cannot do, is be other than what he is. The only thing he can't do, is not be God. This would include things like saying, God is by nature just, therefore God can't be unjust, because to be unjust would be to not be God. It would be to become something other than himself. This displays, again, the principle that true freedom is not freedom to do anything whatsoever, rather true freedom is the ability to be the fullness of who you are, or who you are meant to be.
Now, your statement
For each atomic, indivisible event E in the decision making process, either E is determined, or E is not determined. So "free will" and "random" have to fit together in the same category.
This is really the crux of the issue here. You allow only two categories. Predetermined, and random. As long as you assume this to be true from the beginning, you can never come to any other conclusion.
However, this statement, that there are only these two categories, is not itself provable. It must be taken as self-evident. In order to prove it, you would have to prove that all other categories are impossible.
What I am suggesting to you is that there is a third category that exists between "predetermined" and "random" For lack of a better term I will just call this category "will".
I assert that will is not predetermined because when faced with any given choice, will has the real power to choose any of the available options. This power is real and not apparent only.
I also assert that will is not random because it usually has an end goal in mind and because it makes reference to what is known by the intellect in order to decide between the options available.
I admit that the reference that will makes to intellect, as well as the usual existence of an end goal can be conceived of as influences upon the will, but I deny that those influences constitute complete compulsion or restraint, such that the choice between options is not real but only apparent.
I here make reference back to the three principles I established above regarding freedom. It is possible that the degree of influence may very, meaning we have more or less freedom in any given individual circumstance. It is also possible that any given individual may have become so constrained as to no longer be truly free to make said choices. This however, is a condition of the individual, not of the human nature.
Now, I don't think I can prove conclusively that this third category exists. However, I don't think I have to prove that. All I have to show is that it is possible. The real frame of the question begins with the human lived experience of free will. This lived experience is the basic assumption. In order to deny this lived experience, you have to prove that the idea as we experience it is impossible, and therefore our experience must be concluded to be an illusion. I don't have to prove that our experience is real, I only have to show that it is possible.
You have already stated that you think it is impossible because there are only two categories "predetermined" and "random".
I am asserting that your claim of two categories is not self-evidently true and that there is can possibly be a third category in between the two.
Thus you would have to show that this third category is itself impossible.
I outline all of that because that is how the logical argument and philosophical inquiry would work formally. I fully understand that you are not beholden to any of that to continue to deny my view and believe your own.
It may simply come down to you believing that two categories is self-evident and me believing that it isn't.
This is one of the reasons that logic can't give the kind of certainty that math does. Logic relies upon premises and eventually all premises must rely upon "first principles" or things that are simply self-evidently true. Any logical argument can be "defeated" by simply denying it's premises. It's just a matter of how ridiculous or reasonable it is to deny the premise in question.
For the record, I also have theological reasons why I believe in the existence of free will.
The first is that God must have free will.
If we look back to your argument that predetermined relies upon the existence of pre-existing conditions. Another way of saying this is that the cause of a choice predetermines the choice.
Yet God is, by definition uncaused. There is no pre-existing state.
You could argue that there are "underlying rules" based upon what I said earlier about God's nature. You could conceive of his nature as "rules" but I think this would actually be a serious mistake because rules imply restriction and God's nature is not restriction in the sense that we think of it. It isn't limitation. He is infinite. Our nature places limits upon us, but God's nature does not.
This would have to mean that God acts randomly and "wills" randomly, if your two category view is accurate. However, I think it is also self-evident that God acts with purpose and knowledge, therefore his actions cannot be called random.
If God exists, I think free will must exist.
I further believe that we have free will because we are made in the image and likeness of God. I believe it is the very nature of personhood to have will. Thus I think all personal beings, including God, humans, and angelic beings, have free will.
As for going in circles - I do apologize, I promise I am genuinely trying to understand and doing my best to both see from your perspective as best I can, while also challenging it with what I currently see. I hope that this both helps me change my worldview when it is wrong and hone my communication skills.
This was as much my fault as yours despite my blaming it all on you.
1. This may sound odd, but the more I think about it, I'm not sure I see what the point of free will is. It seems (from the agents taking tests argument) that we can't describe how a universe world of agents without free will would be any different from one with libertarian agents. If we can't even give a mental model of how those universes would be different, then why would I even want free will? It seems like the only effect is that it gives any gods moral justification for eternal reward/punishment.
The point of free will is love. Love, by definition, must be free or it is not love. Love can't be compelled, it is by definition a choice. This can get confused somewhat because our culture views love as a feeling that seeks and finds self-fulfillment and self-gratification in someone else. This is a false view of love. The real definition of love is to freely give yourself for the good of the other.
I further believe that freedom of will is essential to what it means to be a personal being. I don't think personhood can fully exist without free will. Personally I think this is why hell exists. From God's point of view of pure divine Love, the worst possible thing that can ever happen to a person is to become less than a person. To remove a person's free will would be to destroy them as a person, thus hell must exist because the freedom of those who choose against God must be honored as much as it is possible.