The madman's explanation of a thing is always complete, and often in a purely rational sense satisfactory. Or, to speak more strictly, the insane explanation, if not conclusive, is at least unanswerable; this may be observed specially in the two or three commonest kinds of madness. If a man says (for instance) that men have a conspiracy against him, you cannot dispute it except by saying that all the men deny that they are conspirators; which is exactly what conspirators would do. His explanation covers the facts as much as yours. Or if a man says that he is the rightful King of England, it is no complete answer to say that the existing authorities call him mad; for if he were King of England that might be the wisest thing for the existing authorities to do. Or if a man says that he is Jesus Christ, it is no answer to tell him that the world denies his divinity; for the world denied Christ's.
Nevertheless he is wrong. But if we attempt to trace his error in exact terms, we shall not find it quite so easy as we had supposed. Perhaps the nearest we can get to expressing it is to say this: that his mind moves in a perfect but narrow circle. A small circle is quite as infinite as a large circle; but, though it is quite as infinite, it is not so large. In the same way the insane explanation is quite as complete as the sane one, but it is not so large. [...] There is such a thing as a narrow universality; there is such a thing as a small and cramped eternity; you may see it in many modern religions. [...] If we could express our deepest feelings of protest and appeal against this obsession, I suppose we should say something like this: "Oh, I admit that you have your case and have it by heart, and that many things do fit into other things as you say. I admit that your explanation explains a great deal; but what a great deal it leaves out! Are there no other stories in the world except yours; and are all men busy with your business? [...] But how much happier you would be if you only knew that these people cared nothing about you! [...] You would break out of this tiny and tawdry theatre in which your own little plot is always being played, and you would find yourself under a freer sky, in a street full of splendid strangers." Or suppose it were the second case of madness, that of a man who claims the crown, your impulse would be to answer, "All right! Perhaps you know that you are the King of England; but why do you care? Make one magnificent effort and you will be a human being and look down on all the kings of the earth." Or it might be the third case, of the madman who called himself Christ. If we said what we felt, we should say, "So you are the Creator and Redeemer of the world: but what a small world it must be! What a little heaven you must inhabit, with angels no bigger than butterflies! How sad it must be to be God; and an inadequate God! Is there really no life fuller and no love more marvelous than yours; and is it really in your small and painful pity that all flesh must put its faith? How much happier you would be, how much more of you there would be, if the hammer of a higher God could smash your small cosmos, scattering the stars like spangles, and leave you in the open, free like other men to look up as well as down!"
Chesterton knew that logic depends on assumptions, and that perfectly logical arguments can still be flawed if the assumptions leave too much of the ordinary and sane and beautiful out of the picture. And as he showed, the best way to lure a man out of a crazy but internally consistent theory is to offer him a glimpse of sanity that his human nature longs for, but that is incompatible with his theory. He must be convinced that there is something more important to him than his theory. And the process of going about this rescue operation has very little to do with rational debate, but is rather more of a psychological battle to forcibly expose the crazy ideas that the man has taken for granted for too long, the crazy ideas that has made him blind.
The character Innocent Smith from Chesterton's novel Manalive used this method to jolt nihilists, communists and other sad, mad, but perfectly logical people back into sanity. Speaking of Manalive, there's a movie adaptation coming out this year, starring Catholic apologist Mark Shea as Innocent. Watch the trailer!