In Lewis' time, the dominant view of life was what we call Modernism, which placed great confidence in reason, the scientific method and rational arguments. We can also call this view Rationalism.
There are four basic intellectual positions about the relationship between faith and reason( R stands for reason and F stands for faith):
1. R – F = M (modernism or rationalism).
2. F – R = F (fideism or faith-ism)
3. – F – R = P (postmodernism)
4. F + R = C (classical approach)
Lewis took the classical approach: faith plus reason. He definitely saw a place for reason in the Christian faith. He believed there was enough evidence for Christ to lead to the psychological exclusion of doubt, but not the logical exclusion of dispute. While he maintained a place for reason, he was not a modernist. He came to reject rationalism.
The dogmatic rationalist/modernist tries to assert absolutely that "there is no God," which is a universal negative statement. How could anyone go about proving that something does not exist? To know this, you would need to know everything (just like saying that there is no gold in Tasmania would require you to go through every square metre of earth before you could prove your assertion). If there was one thing you did not know, that one thing might be God. The dogmatic assertion that "there is no God" is not only not provable, it is also arrogant. Sometimes a deficiency of argument is covered by dogmatic assertions.
Reason can help to eliminate some of the obstacles to faith, although it can't deal with all of them. Doubt is often more emotional and spiritual than it is intellectual. Our moods can change whatever view our reason has taken. Life is a cosmic balancing act between faith and reason.
For some excellent additional resources, see my post on Apologetics.
[Summarised from Chapter 6 of Art Lindsley's C.S. Lewis' Case for Christ]