As I mentioned before, the logical problem of evil purports to show a logical inconsistency between the existence of God and the existence of evil.  Prominent atheist J.L. Mackie formulated the argument like this:

"In its simplest form the problem is this:  God is omnipotent; God is wholly good; yet evil exists.  There seems to be some contradiction between these three propositions, so that if any two of them were true the third would be false.  But at the same time all three are essential parts of most theological positions; the theologian, it seems, at once must adhere and cannot consistently adhere to all three."

Given the three propositions here the contradiction is not quite explicit, so Mackie continues: 

"...the contradiction does not arise immediately; to show it we need some additional premises...these additional principles are that good is opposed to evil, in such a way that a good thing always eliminates evil as far as it can, and that there are no limits to what an omnipotent thing can do.  From these it follows that a good omnipotent thing eliminates evil completely, and then the propositions that a good omnipotent thing exists, and that evil exists, are incompatible." 

If these propositions are logically inconsistent then at least one of them has to be denied.  Of course, the theist is not going to deny the existence of evil so he is apparently caught in a logical dilemma:  deny God's omnibenevolence or deny God's omnipotence (or worse, deny both!).  The atheist concludes that since evil and God are incompatible, God cannot exist.

Now, here is the strength of the deductive problem of evil:  its logical form.  If the argument succeeds, the conclusion is necessarily true.  In other words, it is impossible for it to be false and that means "game over" for the theist.  So, there's a lot riding on this argument for the theist, as well as the atheist.  However, its strength is also its achilles' heel.  Why?  In order to defeat the argument, all the theist has to do is show that these propositions are logically possible.  The theist doesn't even have to demonstrate that the logical possibility is actually true--the mere logical possibility defeats the argument.

So, given this form of the problem of evil, the theist's task is to show the logical compatibility of God and evil.  But how?