Lewis defines the problem of pain like this: "If God were good, He would wish to make His creatures perfectly happy, and if God were almighty He would be able to do what He wished. But the creatures are not happy. Therefore God lacks either goodness, or power, or both." I think that Lewis believed that he had a solution to this problem, but he admits further on in the same paragraph that "if the popular meanings attached to these words (speaking of 'good', 'almighty' and 'happy') are the best, or the only possible, meanings, then the argument is unanswerable." Normally an attempt to define away the terms of an argument as a means of defeating the argument would annoy the hell out of me, but the way Lewis defines things is interesting, so I'm going to write about it like this isn't a cheap and cowardly tactic.
The first term he addresses is 'almighty;' if I'm reading this right, what he's saying is that an almighty God can't in fact do anything he wishes, but instead has limits. He claims to believe in miracles- I guess he didn't particularly want to be a heretic- but maintains that a physical universe must have certain natural laws that produce a certain amount of suffering, and God is not able to create a universe that doesn't work this way.
"The inexorable "laws of Nature" which operate in defiance of human suffering or desert, which are not turned aside by prayer, seem, at first sight to furnish a strong argument against the goodness and power of God. I am going to submit that not even Omnipotence could create a society of free souls without at the same time creating a relatively independent and "inexorable" Nature."
The proof of this, I think, is supposed to be free will, which requires things to choose among, which requires a physical world, which requires laws of nature, which means the exact laws of nature which we have now and which cause so much suffering.
This approach is very interesting to me. I've heard a lot of people talk about the problem of pain from the angle of goodness, and from the angle of happiness, but I don't think I've ever actually encountered someone who seriously argued that part of the solution to the problem is the idea that God is not actually omnipotent as we understand the word, but rather is constrained by the laws of the universe (but can somehow do miracles anyway, as long as he limits them enough to not actually prove or disprove his existence). It makes sense, but it's not an argument I expect from a Christian, much less C. S. Lewis. If God is subject to scientific laws of nature, you'd expect science to be a big deal, but Lewis is much fonder of weird magical thinking, which he shows in his chapter on the Fall of Man, where he spins a tale that accepts evolution but posits a 'missing link' between pre-humans and modern humans that is essentially super-human: in control of every cell in the body, never dying or ill or in pain, fully one with God and the animals around him and at peace. Like I said, weird and fantastical.
Lewis makes it clear that he believes both that God is limited in power, and that God is something beyond human comprehension beside which humans are vermin. In his chapters on the goodness of God and the wickedness of men, he actually goes into quite a bit of detail about how horrible humans are and how unfortunate it is that modern culture doesn't condemn everything human as utterly worthless and disgusting. These two beliefs don't seem very compatible to me, and if you had asked me before I read this if claiming the omnipotence of God was limited was allowed in orthodox Christianity I would have said it absolutely was not. However, this book is supposed to be one of the best modern works on the problem of pain, so I guess I must be wrong about that. Anyway, Lewis' positions on omnipotence aren't nearly as interesting as his positions on goodness and love, but I think I'm going to write a separate post for that.