1. If the actual world was a world in which God exists, then there would be no suffering of any kind*.
2. There is suffering of some kind.
3. The Actual world is not a world in which God exists.
This is a deductive argument and I am not sure that Peter meant to put it in those terms, so he can forgive me if he ever reads this. I will rephrase it in an inductive way below, so that should cover the bases and one of them should be an adequate summation. The deductive form can be dispatched rather quickly. The argument assumes that
4. God would have no other goods or reasons for allowing (or authoring) evil in any world he creates.
The problem with this move is that this premise is just assumed. All the theists needs to do is simply note that since God is infinite in knowledge and most likely has reasons that we cannot fathom, we are within our rights to posit:
5. God has a perfectly good reason for allowing the evil and suffering that he in fact does.
These reasons could be anything from libertarian freedom if a theist is so inclined to that view to giving an example of some other good that would be an overriding reason for allowing some form of evil in the world. The New Testament actually gives us a pretty good reason for God allowing evil, and that reason was God's plan to save some through his son Jesus Christ. It doesn't matter if someone accepts this premise, what matters is that there is a logical possibility that there is some good that would override God's preventing evil. So the deductive form of the argument fails. However, what I think Peter was getting at was something like the following.
Inductively considered, there is so much evil in the world that God could not have a good reason for allowing the amount and intensity of evil in his creation. Since, there probably isn't any good reason for allowing the amount of pain and suffering in the world, then probably God does not exist.
This argument makes a much weaker claim since it is an inductive argument. It gives us probable premises and probable conclusions. My response to it is similar to the response I gave above, but with a twist. My first point is that we are not in any type of position to know what kinds of reasons a being such as God would have. I can also think of numerous stories that are consistent with reality that undercut the idea that God probably doesn't exist. For example, if the world is Christianity says it is, then we live in a world of morally responsible creatures. God created this world with the purpose of glorifying his son by saving those who rebelled against his law. This was in order to bring glory to the Father and the Son, and this is an ultimate good. This theistic world can be empirically equivalent to an atheistic scenario for the limited knowers that exist in either world. So, evil events E1....En could exist in empirically equivalent worlds W1 and W2. The creatures in both worlds are limited in knowledge, but the weight of the argument by itself only gets us to a .5 probability of God not existing. Given this, we would need to look at different arguments that seek to prove or disprove God's existence. This assumes that all theistic arguments are probabilistic in nature, which is highly debatable.
For these reasons it doesn't seem that the problem of evil should move us in either direction for or against the existence of God.
Peter responded to me later on and noted that we can have three possibilities: 1) There is a world with evil and no second order goods. Second order goods simply being goods that are only found in a world with some evils in it. 2) A world with some evils and second-order goods. 3) A world with no evil and no second order goods.
Peter thinks that world 3 would be better than world 2, but I have to wonder why. I have a conflicting intuition about this. I can give an everyday example. So, it may be a good state of affairs if someone provided for all of my needs all of my life, I never suffer want and I have absolutely no pain when I die. I live in what I called on facebook Lupu Land. I am a happy person with little to no difficulties in life. Call this World 3 person Bill. Then there is World 2 Person lets call him Jake. Jake has a life that is a mixture of good and evil. However, there are goods in World 2 that cannot exist in World 3. So, Jakes world is a world riven by some natural disaster. Jake shows great heroism and self-sacrifice by saving 1,000 people. Now World 2 is equivalent to our world in the amount of goods and pain and suffering. It also has these second order goods, which World 3 does not and cannot have. The people of world 2 are also made of a tougher stuff than world 3. In World 2 people require virtue because things are scarcer, and a little harder than world 3. So people must learn to be good no matter what the circumstances, and they have options to do good or evil. World 3 would not have such people of high moral integrity. It would seem to me that World 2 despite its evils is better than World 3, because there are goods that override material comforts.
* Because a perfectly good being would supposedly not allow evil.