A redated post.
C. S. Lewis lived before computers became the major force that they are now. I think he would not be impressed by the argument that
1) We know that computers are purely material systems.
2) We know that computers reason.
3) Therefore we know that material systems are capable of reasoning.
The reason he would not be impressed is that while computers do the "ratio" part of rational inference very well, the "intellectus" aspect is not to be found in the computer system itself, but is rather a "put in" by human programmers and builders.
I presented Lewis description of the reasoning process in another post, about what he presents in "Why I am Not a Pacifist." It puzzles me somewhat that Lewis didn't spell out exactly what he thought was involved in a rational inference when we was using rational inference to attack naturalism.
The following is from Peter Williams' essay presenting an argument from mind against materialism. See the Lewis quote, which connects to footnote 35 below.
A computer can mimic certain aspects of what scholastic philosophy dubbed "the third act of the mind"  ; that is "reasoning, calculating."  This "third act" is the whole of what people today tend to mean by "reason", and this corresponds to the old French "Raisoner", meaning "to think connectedly or logically".  We can define "the third act of the mind", reason in the French sense, as:
‘The manipulation in thought of beliefs and premises according to the principles of logic, by virtue of which they may be seen in their logical connections, and conclusions may be reached.’
Raisoner is subsumed under the broader Latin definition of "reason" (from the Latin ‘ratio, - on’, meaning "reckoning, judgement, understanding. . ."  ), which corresponds to the scholastic taxonomy of three "acts of the mind":
a) "simple apprehension",
b) judgement, and
c) reasoning [i.e. raisoner]. 
It is the first act of the mind that constitutes intellectus: "intellect (intelligere) is the simple (i.e. indivisible, uncompounded) grasp of an intelligible truth, whereas reasoning (ratiocinari) is the progression towards an intelligible truth by going from one understood (intellecto) point to another."  Thus:
"We are enjoying intellectus when we ‘just see’ a self-evident [basic] truth; we are exercising ratio when we proceed step by step to prove a truth which is not self-evident. A cognitive life in which all truth can be simply ‘seen’ would be the life of an intelligentia, an angel. A life of unmitigated ratio where nothing was simply ‘seen’ and all had to be proved, would presumably be impossible; for nothing can be proved if nothing is self-evident. Man’s mental life is spent labouriously connecting those frequent, but momentary, flashes of intelligentia which constitute intelluctus." 
The first act of the mind, simple apprehension or understanding, contains a subset that has been termed the sapiential sense:
"It is our ability to know these indemonstrable but indisputable truths that, for want of a "cleaner" phrase, we call sapiential sense. Sapiential sense is the mind’s ability to "see" the truths that constitute reality, grasp things as they are in themselves. The "seeing" of these truths transcends the scope of the scientific method (which is limited to the data of the senses) and of logic (which is limited to "unpacking" the conclusions already contained in premises). "Knowledge," writes Illtyd Trethowan, "is basically a matter of seeing things. . . arguments, reasoning processes, are of secondary importance and this not only because without direct awareness or apprehension no process of thought could get underway at all, but also because the point of these processes is to promote further apprehensions."’ 
Each act of the mind builds upon and includes the one before. For, "Knowledge supposes a judgement, explicit or implicit."  Judgement involves the "simple apprehension"  of understanding; and reasoning requires judgement, and thus understanding, which includes "apprehension, intellectual intuition, understanding, "seeing", insight, contemplation."  Rational beings are therefore beings capable of employing all three acts of the mind, for "What we cannot understand we cannot believe; and what we cannot believe we cannot know."  I therefore define reason, in its widest, Latin sense, as:
‘The discerning apprehension of truths which may be manipulated according to the principles of logic, by virtue of which they may be seen in their logical connections, and conclusions may be reached.’
Reason is thus: ‘the combined operation of understanding, judgement, and raisoner in search of truth.’ It is my claim that all three acts of the mind are immaterial and that the human mind is therefore more than material.
While a computer manipulates propositions according to the principles of logic, it does not, I suggest, do this "in thought", as is necessary to the possession of the third act of the mind as defined above. Nor does it posses either the first act, "understanding", or the second act, "judgement". In other words, while computers undoubtedly posses part of the abilities of mind, it is my belief that they do not have mind. Thus I do not think a computer can have beliefs, or, consequently, knowledge. In this I agree with John Polkinghorne who writes that, "The human mind is indeed a computer. . . but it is much more than that - we can also "see", or understand.", and thus that, "The exercise of reason is the activity of persons and it cannot be delegated to computers, however cleverly programmed."  This means that it is impossible to view the human mind as nothing but a biological computer.
As Aristotle argued, "Seeing is an act of the eye, but understanding is not an act of our brain. It is an act of our mind – an immaterial element in our makeup that may be related to, but is distinct from, the brain as a material organ." 
Determinism, Free Will and Moral Responsibility
Physicalism implies determinism, in that the mind is seen as being identical with the brain, which is a natural, physical system running according to the laws of nature. As C.S.Lewis wrote:
"If Naturalism is true, every finite thing or event must be (in principle) explicable in terms of the Total System. . . If any one thing should be such that we see in advance the impossibility of ever giving it that kind of explanation, then Naturalism would be in ruins. . . For by Naturalism we mean the doctrine that only Nature – the whole interlocking system – exists. And if that were true, every thing and event would, if we knew enough, be explicable without remainder. . . as a necessary product of the system." 
Reasons to doubt the truth of determinism are therefore also reasons to doubt the truth of naturalism and physicalism.
One reason to doubt determinism (and thus physicalism) is that it causes severe problems for our concepts of morality. It is not up to the stone whether or not it falls to earth if I throw it into the air. Given certain conditions (being thrown into the air, gravity, etc.) the stone will fall back to earth. The stone has no freedom to do anything other than what it is caused to do; its activity is determined by causes over which it has no control. If humans lack free will, then our actions fall into exactly the same category as the action of a falling stone. We would have no freedom to do otherwise than we are caused to do by causes outside of our control (indeed, we would have no ‘control’ at all). If we are thus determined, does it make any sense to retain belief in moral obligation? A moral obligation is something you ought to do, something you should do; but what use is there for concepts like ‘he ought to do this’ and ‘she should do that’ in a world where every human action is a ‘has to do’? 
We face a choice: either to accept determinism and dump moral obligation, or to retain belief in moral obligation and dump determinism. If we dump determinism, then we must also dump naturalism and physicalism, because naturalism and physicalism entail determinism: "It is safe to say that physicalism requires a radical revision of our common-sense notions of freedom, moral obligation, responsibility, and punishment. On the other hand, if these common-sense notions are true, physicalism is false." 
 Peter Kreeft & Ronald Tacelli, Handbook of Christian Apologetics, (Monarch, 1995).
 T.F.Hoad, Dictionary of Etymology. "When ratio is. . . distinguished from intellectus, it is, I take it, very much what we mean by ‘reason’ today; that is, as Johnson defines it, ‘The power by which man deduces one proposition from another, or proceeds from premises to consequences’." – C.S.Lewis, The Discarded Image, (Cambridge), p157-158.
 Peter Kreeft & Ronald Tacelli, op cit.
 Thomas Aquinas, quoted by C.S.Lewis, The Discarded Image, p157.
 C.S.Lewis, The Discarded Image, p157.
 Roy Abraham Varghese, Great Thinkers On Great Questions, (OneWorld), Introduction, p5-6.
 ‘Knowledge’, The Catholic Encyclopaedia @ http://www.knight.org/advent/cathen/08673a.htm
 Peter Kreeft & Ronald Tacelli, op cit.
 ibid: "We just "see" (in a nonvisual sense of the term) that certain things are true, or that one thing follows from another." (Everitt & Fisher, Modern Epistemology, p4)
 Robert Audi, Epistemology – a contemporary introduction, (Routledge), p183.
 John Polkinghorne, Reason and Reality, (SPCK), p10.
 Mortimer J. Adler, Aristotle for Everybody, (Simon & Schuster, 1997), p183-184.
 C.S.Lewis, Miracles, (Fount), p16, my italics.
 The existence of objective moral obligations forms one premise of the moral argument for the existence of God as the only possible source of such obligations, a conclusion that contradicts naturalism.
 Habermas & Moreland, op cit, p60.This is the passage from "Why I am Not a Pacifist"
C. S. Lewis's Description of Rational Inference
VR: Although C. S. Lewis criticized naturalism by arguing that it is inconsistent with the possibility of rational inference, he didn't give the kind of full description of rational inference that he gives in an essay entitled "Why I am Not a Pacifist," which contains no argument against naturalism at all. It is found in The Weight of Glory, p. 34.
"Now any concrete train of reasoning involves three elements: Firstly, there is the reception of facts to reason about. These facts are received either from our own senses, or from the report of other minds; that is, either experience or authority supplies us with our material. But each man’s experience is so limited that the second source is the more usual; of every hundred facts upon which to reason, ninety-nine depend on authority. Secondly, there is the direct, simple act of the mind perceiving self-evident truth, as when we see that if A and B both equal C, then they equal each other. This act I call intuition. Thirdly, there is an art or skill of arranging the facts so as to yield a series of such intuitions, which linked together produce, a proof of the truth of the propositions we are considering. This in a geometrical proof each step is seen by intuition, and to fail to see it is to be not a bad geometrician but an idiot. The skill comes in arranging the material into a series of intuitable “steps”. Failure to do this does not mean idiocy, but only lack of ingenuity or invention. Failure to follow it need not mean idiocy, but either inattention or a defect of memory which forbids us to hold all the intuitions together.”
The power of intuition, the second step, seems to be the most difficult to account for in naturalistic terms.