Does the existence of evil logically contradict the existence of God? According to virtually all philosophers, the answer is no.
There are two different ways that atheists use evil to argue against the existence of God. The first is the logical argument and the second is the evidential argument. The evidential argument claims that evil makes the existence of God unlikely. But the logical argument makes a much stronger claim. In the 20th century, it was advocated by the Australian philosopher and atheist J.L. Mackie. The logical argument states that the existence of God is logically impossible because there is a logical contradiction between God and evil. That is, God and evil are akin to the idea of a married bachelor or a square circle – a sheer contradiction in terms.
This claim was later refuted by the eminent American philosopher, Alvin Plantinga. Plantinga showed that there was no necessary contradiction between the two. (He also acknowledged that refuting the logical problem did not necessarily solve the evidential argument which needs to be treated on its own terms.) The philosophical community, including Mackie, eventually agreed that Plantinga was right. By the early 1980s, Mackie wrote,
Since this defense is formally possible, and its principle involves no real abandonment of our ordinary view of the opposition between good and evil, we can concede that the problem of evil does not, after all, show that the central doctrines of theism are logically inconsistent with one another.
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Video (below): The Existence of Evil Shows that God Exists
Reference
J. L. Mackie, The Miracle of Theism: Arguments for and against the Existence of God (Oxford [Oxfordshire] : New York: Clarendon Press ; Oxford University Press, 1982), 154.