There are many attempts to articulate this objection to theism. Some, like J.L. Mackie, think that the incompatibility is a logical sort: any evil at all, no matter how small, renders God's existence a logical impossibility. Others, like William Rowe, think that the incompatibility is rather evidential: evil of the sorts, quantity and distribution that we find in our world renders God's existence not impossible but rather unlikely.
I have encountered a certain clever response to this objection. It would have us consider whether the atheist can in consistency admit that there is such a thing as evil.
The dialogue format seems appropriate here. "T" is our theist, and "A" our atheist. A has just finished with her rendition of the argument from evil, and T is about to begin her response.
T: So you say that God and evil cannot both exist?
A: Yes, that's right.
T: But you say that evil does in fact exist?
A: Again, that's right. Indeed I think it obvious that there is evil in the world. Examples are legion.
T: But then must you not say that there is some standard of good and evil?
A: Yes, I suppose I must. There is such a standard, and the things we call evil fall short of it.
T: When you say that, you've opened the door to theism.
T will now attempt to prove that a standard by which we distinguish good from evil of necessity leads back to theism. Lewis seems to have had just such a strategy in mind when, in Mere Christiantity, he said this:
My argument against God was that the universe seemed so cruel and unjust. But how had I got this idea of just and unjust? A man does not call a line crooked unless he has some idea of a straight line. What was I comparing this universe with when I called it unjust? ... Of course I could have given up my idea of justice by saying that it was nothing but a private idea of my own. But if I did that, then my argument against God collapsed too--for the argument depended on saying that the world was really unjust, not simply that it did not happen to please my private fancies. Thus in the very act of trying to prove that God did not exist--in other words, that the whole of reality was senseless--I found I was forced to assume that one part of reality--namely my idea of justice--was full of sense.I do not doubt that some who press the problem of evil would admit to the existence of evil and thus to a standard of good and evil. But they need not do so. One can reformulate the argument from evil so that it no longer assumes that evil exists. This may seem paradoxical at first, but let me show you how it's done. The dialogue begins as it did before.
T: So you say that God and evil cannot both exist?
A: Yes, that's right.
T: But you say that evil does in fact exist?
A: No, you've misunderstood. All that I've said is that the theist must say that evil does in fact exist.
T: But don't you have an opinion on the matter?
A: I do, but my opinion is irrelevant to my objection. Here, let me map out the argument for you. (A stands up and walks over to a nearby chalkboard.) I'll write out the objection to make sure it's clear. (A numbers his propositions like a good analytic.)
1. There are such things as murder, rape, cancer, hydrocephaly, etc.
2. The theist must say that these things are evil, or to put it in conditional form: if theism is true, murder, rape and all the rest are evil.
3. But if murder is evil, then given that murder is all too real, there is such a thing as evil.
4. If evil exists, God cannot exist. (After he writes 4, A turns to T and asks him to recall the argument he gave for 4. He reminds A that it has to do with the impossibility that a perfect God would allow evil to exist.)
5. Collect together 2, 3 and 4 and we reach this sub-conclusion: if theism is true, God cannot exist.
6. But this leads us straight into logical paradox. For surely we must say that if theism is true, God must exist. Indeed theism just is the claim that there exists a God.
7. Thus if theism is true, we must say both that God exists and that God does not exist.
8. We find, then, that theism entails a contradiction - the contradiction that there both is and is not a God.
9. Hence we must say that theism cannot be true.
(A returns to his seat and begins to talk about what he's just written.)
A: So you see, then, that in my argument I do not say that there's such a thing as evil. Rather all that I say is that there are such things as murder and cancer and that the theist must say that these things are evil. In short, it is not I who say that there's evil. Rather it's you who says that there's evil, and in my argument all that I do is report that belief of yours. After I report it, I attempt to isolate an inconsistency in your belief-set.
I'll break off the dialogue at this point. I can think of nothing for T to say but to grant A's point. The upshot, of course, is that, when the atheist presses the problem of evil, she needn't leave the door open to theism (at least not in the way Lewis thought). In particular, she need not herself assume that there's any such thing as evil. She reports a belief in evil. She does not assent to it. Moreover, let us be careful about what T has attemtped to do. She has not attempted to show that there's no God. Rather she has attempted to show that there's deep inconsistency in the Christian world-view. The Christian's assent to God's existence and to the existence of evil, says T, shows that the Christian's world-view is deeply inconsistent. In this context, A offers no suggestions about how to remove that inconsistency.