1. If evil is a problem, then God exists.
2. Evil is a problem.
3. Therefore, God exists.
The logical form of the argument is modus ponens:
1. If P then Q.
2. P.
3. Therefore, Q.
Two key words in the argument that need definition are "problem" and "God."
By "problem" I mean the conviction that evil "could and ought to be otherwise," a judgment that the existence of evil is "wrong," that the cosmos is "broken," that evil mars reality as a "flaw."
By "God" I mean "a power that could make reality otherwise than what it is."
Okay, how serious am I expecting you to take this argument? I'd say semi-seriously.
On the non-serious side, I don't think this is very convincing argument or "proof" for the existence of God.
But on the serious side, the argument highlights how our felt experience of evil is that evil is a problem, a flaw with the design of the cosmos, and how that experience presupposes a power (God) that could, and should, make things otherwise.
Of course, not everyone experiences evil as a problem or design flaw. Suffering, pain, and evil could be taken as brute facts of existence. And brute facts cannot be "problems." Facts cannot be flaws. Evil might be tragic and sad, and we might wish it were otherwise, but since it can't be otherwise there's nothing "wrong" or "off" here. There's sadness, but no problem.
But my sense is that most people experience evil as a problem. We feel something has gone "wrong" when a child dies of cancer. We feel that this tragedy could have and should have been otherwise. This death revealed that something is broken in the cosmos, something that should be fixed and rectified. Sure, perhaps a logical interrogation of our grief would reveal it to be sentimental and superstitious. But I think a lot of us would go with our gut over logic in this instance as the more truthful. Evil really is a problem.
Therefore God exists.