I know plenty of people, including Christians and theologians, who do not take Lewis seriously as a thinker. I think this is a mistake. He is not a major philosopher or theologian like, say, Lonergan, or even like his friend Austin Farrer; it would have appalled him to be considered one. But he is one of the preeminent minds of the 20th century when it comes to what used to be called "men of letters," a category which in the long run may be the more significant. (It is not the same as "intellectuals.")
The "Argument From Reason," is not, of course, Lewis’ alone; a number of other thinkers, notably Plantinga, adapted it. It seems to me to be one of those convenient litmus tests for “kinds of thinkers;” nobody tends to be neutral about it. You either find it compelling (at least in a strange sort of way, maybe like the ontological proof of God), or you find it makes you squirm with impatience -- how could anyone ever find such an “argument” persuasive?! This makes it a either a conversation-stopper or an ideal conversation-starter, depending.
The argument, in very rough, indeed caricaturish, outline, is as follows: naturalism is the claim that nothing but natural processes exist and occur. These processes are all cause-and-effect processes; indeed, according to naturalism, there is no other kind of process. Such processes, being exhaustive of everything, obviously per hypothesis include the human mind and its conclusions, whether false or true. But this entails, then, that any true conclusions must have been occasioned by cause-&-effect, and in fact by “causes” that are not strictly what we recognize as “reasons” at all. We can, in fact, not have reasons for believing anything at all, including naturalism, if every “reason” reduces to a cause in the ordinary sense.
Anscombe’s case against Lewis has a decidedly “analytic” flavor to it (unsurprisingly), even an “ordinary-language” flavor, as, e.g., her argument that a reason is not what produces a belief but is rather “what is elicited from someone whom we ask to explain himself.” This reads like everyday common-sense to someone who has been steeped in Wittgenstein, and the first several times I encountered it I breezed right past, but in fact it is starkly implausible and surely gets the phenomenology of insight very skew. Nonetheless, the story of Anscombe’s “defeat” of Lewis became a kind of bit of received wisdom, an anecdote substituting for an argument, and has played a role both in the general dismissal of the “Argument from Reason,” and its adoption by special interests, like Plantinga’s -- widely perceived as rear-guard actions in a losing defense of Christianity against the inevitable progress of science. I have some sympathy for the underdog in that scenario, but framed in those terms, it will never do.
I have come to regard the Lewis-Anscombe debate as a late and minor skirmish in a war that was already over, had in fact been over for some time. Lewis’ intellect was shaped, as he said himself, out of the confluence of a number of factors: leaving aside aesthetic and what we would (but he would not) have called existential concerns, these were Scottish common-sensism (derived from Reid) and the Idealism of Bradley, Green, Bosanquet, and so on. Neither of these were the atmosphere of the Socratic Club at Cambridge. The atmosphere that made Anscombe’s arguments so persuasive, so that the verdict of the audience was that Lewis needed to make his argument “more rigorously analytical,” was one that had been made by Russell and Moore, and by Wittgenstein. This does not mean that those audience members simply concluded that Anscombe was right “because they were analytical philosophers” -- as though the weight of the arguments themselves were so much sizzle. To say this would be to commit the fallacy Lewis memorably described as Bulverism. But one may avoid Bulverism and still acknowledge that fashions matter in the history of thinking; that conclusion which look altogether inevitable in one context look starkly implausible in another; and that adjudicating between these contexts is no simple matter of “merely thinking honestly.” We think in contexts, not outside of them.
In fact, a second or third look at the famous disputes -- mainly between Russell and Bradley -- which re-shaped the philosophical landscape in Britain in the early part of the twentieth century, leaves little room for doubt that the change was not decided on the merits of the arguments alone. This re-evaluation has been undertaken by Stewart Candlish in a really valuable work of scholarship (and not just scholarship), The Russell/Bradley Dispute. From a hundred years’ distance, it is obvious that Bradley was not “defeated” by Russell’s arguments, and that in many cases Russell seriously misconceived what Bradley had said or meant. What really happened, it seems to me, is that philosophers got tired of talking in one way, and were excited and intrigued by the possibilities of talking in another way. This sociological slant does not mean that there is no such thing as genuine philosophical insight to be had. It does not consign us to a maze of relativism. But it does mean we must be more cautious in rejecting out of hand positions to which we are unsympathetic, or at least, assuming that we have the weight of argument on our side when we do so.
Of course, if Lewis is correct, claiming that “the weight of argument” has any bearing on the case at all, may commit us to certain other ramifications -- not just epistemological, but ontological. For a deeper consideration of those claims, I refer you to the Quodlibet series, and its numerous references.