Purpose implies a willful agent.

Willful intent is a decision making process to put certain events into place to establish certain consequences. And all of this assumes a premise/conclusion type of relationship between cause and effect. So I do not believe that the Will of man or God is something that cannot be defined logically.

To go from suggesting that the universe must be logical to suggesting that its "very purpose" is to "demonstrate what is logical" is a big step.

"Purpose" is still another word that describes the relationship between events and their consequences. For instance, the purpose of putting a pan of water on the stove is to bring it to boil. This is other words for saying, "IF you put a pan of water on the stove, THEN it will come to boil." And you can see the premise and conclusion in this last sentence. The word "purpose" suggests an intelligence who knows what will happen if you do something and does it because it intends to effect that consequence. Still, I think that the principle applies. I could have just as easily have used the words "The universe WILL demonstrate what is logical" without lose of generality. For the universe goes from state to state without display of the general principles employed. The general principles are not distinguished in that single transition of state. In order to demonstrate the general principles that determine the universe, it requires memory to record the relationship between many shifts in state. Only then can generalities be determined. Then if there exists entities that exist only if they use the correct generalities, the correct general principles that govern the universe will be demonstrated by the existence of those who survive.

You seem to be basing your conclusion that the universe is not unreasonable on the fact that reasonable debate would not be possible in an unreasonable universe. Let's grant that point for a second. All that proves is that the universe doesn't violate the laws of reason. For example a statement about the universe cannot be both true and untrue at the same time. It does not prove that the universe is capable of reason or that it was created for a purpose.

So you are saying that I would be using the word "reason" equivocally - one definition of reason refers to the subject as a whole, the other definition refers to the action or process of reasoning. I do not necessarily mean to say that the act of reasoning by which events are determined proves a conscious being that is doing the reasoning. The action of reason that determines how one event will proceed from another event may be instantaneous and automatic without conscious effort. Reason (or logical consistency) may simply be the underlying force of the universe. And so when used in this sense, I don't think that I am using the word "reason" (or "demonstrate" for that matter) equivocally. For I can refer to any other force in the universe, say gravity, both as a general concept and as an action that produces events.

But perhaps you are having more trouble with whether I am equivocally using the phrase, "the universe will demonstrate the principles of logic." One meaning is that the principles of logic are demonstrated by the events in the universe simply because the events of the universe are merely consistent with logic. Another meaning is that the general principles will be abstracted from a concrete universe (and this in the form of human beings). This is mathematically equivalent to asking whether the axioms of any system of logic can be derive from the theorems. Intuitively, it seems necessary that the axiom are implied by the theorems because you would not even be able to write the theorems if the axiom did not exist. You must use the rules of writing theorems to write a theorem.

In any event, if humans are able to abstract the general principles of cause and effect, premise and conclusion, and use them to maintain their existence, then since humans were produced using the concrete material universe, the concrete universe DOES and WILL abstract the principles of the universe. For the universe produced humans that do abstract. Furthermore, if humans only exist if they are able to understand (through abstraction) the principles of cause and effect, premise and conclusion, then their existence is determined by the fundamental principles of logic whose fundamental variables are either true or false. They exist if their understanding is true, they cease to exist if their understanding is false so that their existence is logically equivalent to the validity of their understanding of the logical principles that they follow.

But your argument was that the proof of that fact could be demonstrated in the inability of anyone to prove that the universe was illogical. That fact doesn't prove anything. You were presumably presenting a complete argument based on that false conclusion.

No, I did not mean that the logical consistency of the universe was proven by the inability of humans to prove otherwise but by the inability of logic itself to prove otherwise. The principles of logic cannot prove that anything does not comply with the rule of logic. For it is only capable of defining entities that do comply with the rules of logic.

Many people have argued that reality might transcend any finite conceptualization of it.

They are wasting their time, for it is not within the ability of any logical system to suggest that something does not comply with that system. For any system of logic only deals with variables that can be defined in that system. You cannot define any entity that does not comply with the rules of that system. So they could never formally prove that a statement is not logical.