Well, we've still got a significant problem: that word, "evil."
In order to discuss what "evil" is, or why it exists (assuming it does), or who is responsible for it (assuming anybody is), we all have to know already what "evil" actually IS. Or rather, we have to take for granted that we have common ground about what evil looks like, how to define it, and how to recognize it.
As for logic, logic is a procedure, like mathematics is a procedure. You can plug any sorts of assumptions into it, and logic will work, provided the rules of logic are followed. But one of the fundamental rules of logic is "no equivocation of terms." That means that when discussing or debating something, the key terms being discussed must retain a singular, clear and stable meaning. To remain logical, we have to use the same meaning for the words, and keep that meaning stable throughout the whole discussion. Failure to achieve this is called a "fallacy of amphiboly," and always results in illogic.
In the case of employing logic to discuss evil, that means that logic itself tells us we have to have a singular, clear, common and stable definition of "evil." If you and I don't have a common understanding on a key term, we'll lapse into a fallacy of amphiboly, almost certainly, and end up discussing very different things, and miss each other like ships in the night.
But here's the problem that remains for us: Theism tells us what "evil" is. (We can agree with the Theistic definitions, or disagree, but Theism certainly has positions on moral terms like "evil.") However, skepticism -- Atheism in particular, Materialism, Physicalism, and all forms of moral subjectivism -- has no grounds for any definition of evil. For such worldviews, "evil" must mean something like, "phenomena I don't feel good about," or "things I don't like," and no more than that. And the problem, of course, is that not only can such a definition not show that anything is "evil" or "horrible" or "bad" in any objective and stable sense, but that this definition is also not at all the definition assumed by the Theist, who does not think that "evil" can be defined on something so thin as a personal feeling.
So the Theist and the subjectivist skeptic have no common concept of evil from which to start. No logical syllogisms they work out will then be logically valid and correct. They'll fall afoul of the fallacy of amphiboly.
But if you can propose a way a subjectivist or skeptic can assert an objective conception of "evil," then I think logic will serve us admirably, going forward. At the moment, however, we do not have that established.
