I'm going to redate the old post, but I'm fixing the argument from evil to leave out the stuff about the best of all possible worlds.
The title here suggests that Arminians (or maybe universalists) can solve the problem of evil, and maybe that's something I don't want to imply. However, I am going to argue that while the theist has some hope of providing a rebuttal to the argument from evil if Calvinism is false, the theist who is as Calvinist is thoroughly embarrassed in the face of the atheist argument.
Let's look at the atheist argument from evil. The argument goes as follows:
1. God, if God exists, is omniscient, omnipotent, and perfectly good.
2. If there is a God, then there is no unnecessary evil.
3. But there is unnecessary evil.
4. Therefore God does not exist.
Now a lot of responses to the problem of evil employ two themes. One of those themes is that many evils in the world are not caused by God, but are the result of God's allowing creatures to act freely. If God makes us free to commit murder or not to commit murder, then God cannot guarantee that we not commit murder. Second, some things that seem evil from a temporal, present-day perspective may not be evils from an eternal perspective. So, for example, it may seem to me to be an evil that I have a certain health problem, but from God may see that there are elements of character that will improve within me if I suffer from this health problem.
The problem with Calvinism is that on the Calvinistic view God sovereignly determines the outcome of every action. And there are situations which persevere into eternity which very clearly could have been better. In particular, "Smith's going to hell" is a situation which goes preseveres into eternity and is not going to get better.
Consider what philosopher Douglas Jesseph calls "The World of Mr. Rogers."
In the world of Mr. Rogers, it's all a big happy neighborhood and everyone does what is right, and then go to heaven when they die. This world is obviously a better world than this one. Just ask anyone who has gone to hell and see if they wouldn't prefer the World of Mr. Rogers.
Do creatures in the World of Mr. Rogers act freely? If you accept the Compatibilist theory of Free Will, then they do. The Compatibilist points out that there is a difference between doing what you want to do, and doing what you don't want to do. You act unfreely if you are dragged in chains back to your jail cell. You act freely if you are listening to your favorite CD, playing your favorite game, watching your favorite movie, etc. When your actions and your desires match up, you are acting freely. If your actions and desires don't match, then you aren't.
Now suppose you have the desires that you are caused to have. God arranges it so that you always want to do the right thing. God controls your desires and then gives you the freedom, that is the power, to act out those desires.
Given this theory of free will, it is easy to see that God could still give us free will and nevertheless still be able to create the World of Mr. Rogers, where no one sins and no one goes to hell. If Calvinism is true, then that is exactly the kind of free will that God has granted us. Calvinists will often say that God has every right to punish us for our sins because, after all, it is not the case that we wanted to do the right thing and were forced against our will to do the wrong thing. Rather, we did what we wanted to do, even though we were predestined to do it.
Is the World of Mr. Rogers boring? Well, if it's boring, then heaven will be boring too, because heaven is supposed to be a sin-free zone.
The simple fact is that if Calvinism is true, then God could have created the World of Mr. Rogers, but sovereignly chose not to. Why?
At this point it is possible to now appeal to human limitations, either limitations in human knowledge or in human goodness. Even though we can't see that this world is better than the WMR, it really is better, even though some people are damned in this world and no one is damned in WMR.
I think these arguments from the limits of our knowledge have more force where the final outcome is unknown or inadequately understood. We know the final outcome in both worlds. Everyone is happy in the WMR and everyone gets saved. Many people suffer in our world and some are lost.
Another way of replying is to present a version of Paul's rebuttal from the Book of Romans, "Who are you, o man, to answer back to God?" Now if this is a version of the argument from the limitations of our knowledge, which I think it is, then it has some value, but not on a Calvinistic scenario. If however, it is a way of simply dismissing the argument from evil, it is a transparently question-begging argument. The AfE questions whether there is a God, that is, a being omnisicent, omnipotent, and perfectly good, to answer back to. We cannot assume that such a being exists in order to eliminate the question as to whether or not an OOP being exists.
But why would God want to give us any kind of free will other than the kind of free will that the compatibilist (and the Calvinist) is prepared to admit? The incompatiblist holds that human beings have the kind of freedom that is incompatible with our acting from a determining cause. If we sin, we could have done otherwise under the actual circumstances.
Consider a Star Trek I once saw. There was a man, who in the show was named Flint, who was born several thousand years BC, whose body was able to regenerate whenever it was damaged, granting him an virtually endless life. What that meant was that, over and over again, he saw his companions and wives die. He ended up on a planet in outer space where he decided to build the perfect companion, an android named Rayna. Rayna could converse with him on any subject imaginable, could be physically affectionate, but there was one problem. Its "love" for Flint was fully and completely determined by Flint's programming, and therefore was deficient as love. So Flint brought the Enterprise and Captain Kirk to the planet so that he could be a rival for Rayna's affections. (Captain Kirk was, in more ways than one, modeled after President John F. Kennedy--JTK, JFK. Say no more.) Anyway, since Rayna was an android, Rayna couldn't choose freely, and so fell over and became deactivated.
If God is love, then isn't there something deficient about love that is fully and completely determined by the one who recevies the love? If this is the case, then there is a good reason why a loving God might choose to give us incompatiblist freedom, even if this freedom results in sin and perhaps even eternal separation from God for some persons.