Dostoevsky depicts a conversation between two brothers: Ivan and Alyosha. Ivan is an atheist (a weak category, perhaps) and he observes all the suffering going on in the world. Alyosha is a Christian and is attempting to bring Ivan closer to the truth that there can be forgiveness. Ivan has this rebellion against God and doesn’t believe there is such a being that will actually forgive the world of these evils. Ivan makes the distinction between the sufferings of children and the sufferings of adults. It’s the adults that Ivan has little sympathy for since they are the only creature that can practice artistic evil. A tiger maims its prey but it would never even think of nailing the prey’s ears or blowing the brains out of another animal, even if it could. He suggests that the adults, even the elderly, should be damned because they have eaten the apple (participated in conscious acts of sin). However, the children have done nothing wrong. Children have just shown up in existence and have no conscious control or recognition of morality. When these children are bayoneted, shot in the face, or receive long tortuous beatings and those who fulfill these acts take joy in it this becomes an artistic evil and is unwarranted evil against the children because they have not eaten the apple.
The argument wasn’t so much of a conclusion that God doesn’t exist; rather, it was Ivan’s way of expressing his desire to “return God’s ticket.” The world is unjust and this was Ivan’s rebellion against God.