Since I am not a philosopher, I asked Dr. David Gordon for his response to the comment, which follows:
“If someone denies the Action Axiom, he is not saying, “No actions are purposeful” He is saying, “There are no actions.” The contradiction he falls into is that his making this statement is an action, so his making the statement shows that there is at least one action. The contradiction is not that his statement shows that there is at least one action that is purposeful, so it is false that no actions are purposeful. “Non-purposeful action” makes no sense. It is true that the contradiction in denying the Action Axiom shows only that there is at least one action, but that isn’t the way we know there is more than one action..
“The statement ‘human action is purposeful’ is a definition, where by a definition is meant an account of the essence of action, not a stipulation. In other words, that’s what action is. We know that this definition is correct by thinking about the nature of action, That is, we realize that there is such a thing as purposeful behavior.”