The Problem of Evil, as most of us know, can be stated like this:
- God is the all-powerful and all-benevolent creator of the universe.
- Evil exists in that universe.
- If God is all-powerful, and yet evil exists, God must not be all-benevolent.
- If God is all-benevolent, and yet evil exists, God must not be all-powerful.
- So, the all-powerful and all-benevolent God does not exist.
The argument is sound provided that you accept its definition of God.
However, the definition depends on a conception of God that is akin to our conception of geometric shapes. For example, if a polygon has three angles, then it is a triangle. If it has some other number of angles, it is not. So too with the definition of God in the argument above: If a being is all-powerful and all-benevolent, then it is God. If the being lacks one or both of these characteristics, then it is not.
However, most theists conceive of God as a person rather than a geometric shape. The former, unlike the latter, can have characteristics that are apparent but not actual. Also, a person has many more than two identifying characteristics. For example, God, in addition to being all-powerful and all-benevolent, can be the creator of the universe, Heaven, and Hell; incorporeal and normally invisible; responsible for anomalous events; unitary or triune; able to incarnate himself; and the reason some people stop overdrinking, to name a few.
As for apparent vs. real characteristics, consider a nonce person we'll call "Greg," a thirty-year-old man who lives in a middle-class neighborhood with a low crime rate. Greg helps little old ladies across the street. People trust him to babysit their children, and he has never violated that trust. He has no criminal record. His neighbors praise him frequently for his cheerful willingness to help them whenever he can. In light of all this, people in his social universe credit him with consistent moral goodness.
That's because the latter people don't know that Greg is a serial killer who abducts people from a neighboring city, takes them to a secret sound-proof room beneath his kitchen floor, and uses the equipment there to process them into canned dog food, which he feeds to his dog. Greg finds his clandestine activity to be gratifying in a repugnant way.
Now imagine that Greg is found out, thanks to a brilliant detective on the local police force. The neighbors are shocked. The neighbors say that Greg is not the consistently good man that they thought he was. But the neighbors do not say that Greg does not exist. They might say that the good Greg they knew did not exist, but they would say so only in a metaphorical sense. They would know that, literally, Greg still exists because Greg is a person and, as such, can have can lack an apparent characteristic, possess most of his defining characteristics, and still be Greg.
If we conceive of God as a person, rather than some mathematical entity, then God can exist even if he lacks one of these two characteristics: a) being all-powerful, and b) being all-benevolent.
Indeed, I once knew someone who lost his children in a natural disaster. He credits God with the strength to carry on after that tragedy but does not believe that God is all-powerful. (Yes, this is real.)
As for a God who is all-powerful but not all-benevolent, such a being would constitute a good theistic explanation of many of the events we all read about in newspapers.