What, exactly qualifies as naturalism? And what did C. S. Lewis think he had refuted?

 In defending the central argument in the third chapter of Miracles, we have to ask exactly what Lewis meant by naturalism when he said that naturalism is self-refuting. David Kyle Johnson thinks there is a standard definition of naturalism--an entity is natural if it is part of the universe, and naturalism is the view that only the universe exists. It has nothing to do with whether mental states are fundamental to the universe or not. Hence an argument that shows that there must be basic mental causes would not be sufficient to refute naturalism, since there are versions of naturalism where this is not denied. 

David Kyle Johnson, "Retiring the Argument from Reason," Philosophia Christ Vol 20, no. 2., (2018). 

But, in fact, the difficulty in defining naturalism is widely noted, and so far as I can tell, there is no standard definition. Alvin Plantinga defines naturalism as the view that there is no God, or no being like God. But what is God-like enough to be a problem for naturalism? 

Rickabaugh and Boras state that naturalism is a thesis about ontology, but is also a thesis about explanation. They note: 

The distinctively scientific mode of explanation is subpersonal and mechanistic.16 To give a mechanistic explanation of some phenomena (for example, change in location) is to cite a property of an object (for example, the mass of a body) together with a natural law (for example, Newton’s inverse-square law) describing how things with that property regularly behave. Such laws describe the most general patterns of variation in nature, based on the inherent tendencies of things. Mechanistic explanations thus tell us what we can expect to happen automatically (deterministically or probabilistically) and, as it were, of its own accord. That is, whether prior events strictly necessitate or fix the chances of a future event, still that event is the automatic result of nonrational causes. Deterministic laws predict a fixed outcome for phenomena that fall under their jurisdiction. Probabilistic laws, by contrast, assign a probability to all possible outcomes, and leave it to chance to resolve which comes to pass. Mechanistic explanations thus tell us that some phenomenon occurs because the state of the universe and the laws of nature necessitate it or make it somewhat likely. Here is our point: the central idea of naturalism (at least with respect to explanation) is that mechanistic explanation is in principle complete, that is, sufficient to explain everything that needs explaining.

16. The account of scientific explanation in this paragraph and the next follows Richard Swinburne, The Existence of God, 2nd ed. (New York: Oxford University Press, 2004), 26–35, which is in turn based on the classic exposition in C. G. Hempel and P. Oppenheim, “Studies in the Logic of Explanation,” Philosophy of Science 14 (1948): 135–7

Brandon Rickabaugh and Todd Boras, "The Argument from Reason and Mental Causal Drainage: A Reply to Peter van Inwagen," Philosophia Christi vol 19, no. 2, 384. (2017). 

Lewis presents naturalism and supernaturalism as two options, so based on that you might think that he would be happy with Johnson's definition. However, Lewis also has a chapter on "Christianity and Religion" in which he argues against pantheistic views, and there he does not use an argument from reason. These pantheistic views include a position he once held, and the position he adopted once he became persuaded that naturalistic views "leave no room for an adequate theory of knowledge." That is, he became an Absolute Idealist. According to Absolute idealism, God is not distinct from the universe, and so by Johnson's definition, it qualifies as a form of naturalism. 

In any event, in the name of naturalism many people hold a position that says that the base level is mechanistic (in the sense described), the base level is causally closed, and whatever else there is supervenes on the base level. Most of them call the base level physics, which makes them physicalists. If these positions are refuted by the argument from reason, then it is hard to call the argument from reason a failure, even if it leaves space for people to depart from that position without embracing theism. After all, that is just what C. S. Lewis did when he was first convinced that naturalism as he understood it did not hold up. As he said "I didn't want there to be a God, I didn't want the universe to be like that."  Oh wait, with a tense change, that's Thomas Nagel. I apologize for the confusion.