Here, I will try to clearly articulate a common argument against the existence of God, the free will defense, and a reply to said defense. I will then argue that if you accept the premises of the argument relating to the nature of God, then you must follow the conclusion that God cannot exist. You will notice that put the word "logical" in the thread title. This simply means the version of the problem of evil being discussed tries to show God is logically incompatible with the existence of evil. In other words, the definition of God cannot coexist with the current (and present) existence of evil because their mutual existence logically contradict each other, much like a square-circle cannot exist.
To start:
1) Evil and suffering exists. (Justificaiton: prima facie true.)
2) God exist, a being who is: (Justification: Assumption)
a)omnipotent (able to do all things logically possible),
b)omniscient (knows all true and false propositions), and
c)omnibelevolent (wills the highest good of the other agent. For example, this
highest good can be achieving a relationship with God and getting into heaven).
3) The good agent wants to avoid as much evil and suffering as possible; the good agent wants avoid all unnecessary evil. For example, if the agent wants to teach someone something important, he strive for the path that causes the least amount of unnecessary suffering. If the agent can either teach me through non-painful tutoring or painful torture, the agent will always choose non-painful tutoring. (Jusitifcation: prima facie true)
4) God wants to avoid all unnecessary evil and suffering in achieving just ends; any unnecessary evil will be stopped by God. (Justification: 2c)
5) There exists no necessary evils in our world. (Jusitifcation: 2a and 2b. God can always teach us through non-painful ways, for example. There is no just end being served in allowing the torture of innocent people)
6) Therefore, God is incompatible with the existence of evil. Because evil clearly exists, God cannot exist. (Justification: All)
The obvious response is to reject Premise 5 and claim that God must respect the free will of agents as a necessary evil to achieve the just end of freedom. Here, I will respond to this claim.
1) God has divine-foreknowledge of the actions of free creatures. In other words, God knows what an agent with free will ultimately chooses before that agent even actually exists. God knows if I will freely rob the bank or refrain from do so, for example. (Justification: God's omniscience).
2) Mackie's world is possible. Mackie's world is a possible world where all free agents choose to do the morally good action with every choice. In other words, no one does anything evil, so evil does not exist. (Justification: prima facie true. Though it seems improbable and odd, it is logically conceivable, so it is, therefore, logically possible. There is no reason to believe such a world is not possible.)
3) God can actualize Mackie's world. (Justification: Premise 1, God's omnipotence and omniscience. God is aware of Mackie's world and create said world, as God can create all logically possible worlds).
4) Therefore, the free will defense does not stand, as the existence of free will does not necessitate evil existing. God could have created a possible world with both free will and no evil.
I admit that Premise 1 of the second argument is questionable. If you believe God cannot know actions that result from free will before they happen, then the argument does not apply to you. However, if you accept this premise, please explain where this Argument from Evil fails.