Logic is a tool that helps us detect contradictions and avoid them. It helps us think and gives us much protection from error and lies. A logical contradiction is no better than a lie. You may as well lie. To argue that your dog is made of steel is illogical. You know it is not steel so to say it is, is to lie. Try not to think logic but feel it too. Don't let it be a cold enterprise.
A God is all good and all powerful.
B Evil exists
Conclusion: God is not real for evil contradicts his goodness or power. An evil or limited being is hardly a God!
Bertrand Russell said that evil might not be a strict logical disproof of God for it may be that there are goods that are so great that it is better that evil and good should both exist than that neither should. He does balance this by asking this, "What if good things only exist to afford opportunities for greater evils?"
This is very future-orientated. If there is no future or if God just stops keeping things in existence in a few minutes he has got his logical disproof of God.
Logic bans you from saying you know what God will or will not do. So far we don't know then if evil refutes a good God by being a logical contradiction.
Christians say if there is no God then we have no reason to call anything good. And therefore call anything evil. For that reason it says no logical argument from evil against God can succeed.
Ahem! That is irrelevant to the validity or otherwise of the logical argument that evil refutes the love of God. A valid argument is one where the conclusion follows from the premises. An argument being valid does not mean the argument is true.
But if evil logically disproves God and if without God we have no grounds to call evil evil then so what?
The two together would refute God as a coherent idea. Such a contradiction only arises if you are trying to defend a notion, God in this case, that does not make any coherent sense in the first place.
Religion bases itself on the notion that there may be a God regardless of evil and and suffering for it focuses on good coming from evil. It thinks that implies a God sees that evil has to be tolerated until it can be destroyed. But the fact is as Moore noted, you do have a reason that has nothing to do with God. You just look and you see what is good. God is not the source of good so you cannot look at him to see what good is. Plus you are assessing him in accordance with how you see good. So while you say God is good what you really mean is, "I decide what good is and he fits my criteria". It is back to you. It should not be about you.
Religion in fact cannot really work out what good and evil are. Christians shamelessly avoid going into exactly what they mean. Ideologies are typically messing around with words and are unclear where there is something to hide. Too often what is called good is mixed up with what good actually does. If good makes you happy that does not mean that goodness is the same as happiness or should make you happy. So if we cannot hit the nail on the head with good we cannot with evil either. No two believers know how to handle a criminal and punish correctly so when they don't know what justice is clearly how can they be clear on love and on goodness?
Once again for a new reason we don't know then if evil refutes a good God by being a logical contradiction.
Evil implies a level of stupidity and incompetence so a God who is evil is by definition not a God. His faults are bigger than him so he is not supreme.
So far we are left being agnostic. It could be that evil logically disproves God or does not. If we leave it there then all we can say is, "Evil might logically show that God cannot exist. Or maybe it does not. We don't know either way."
So here you get a logical argument against saying we should believe in God. That is not the same thing as a logical argument against God.
So would that push us to Agnosticism? Yes. Agnosticism leads to an interesting dilemma when it comes to worship. Worshiping an evil God is wrong, just as falsely accusing a good God of being evil is wrong. It is right to reject worship of an evil God and right to worship a truly good God. Good and evil cannot be compromised or meet halfway. Yet, agnosticism attempts to strike a balance—it withholds worship from a God who might be good and deserving, while simultaneously opposing worship out of fear that God might be evil. This contradictory stance sends a mixed message and cannot truly be considered good. By definition, good does not settle or compromise with evil.
If God exists and can only be evil and that is absurd then the worship of God is nonsense. We have proven that if God is not logically disproved by evil committing worship and yourself to that God is illogical. If we start to say God would be AN EVIL, that is a different claim from saying he IS EVIL. A kindly doctor who only means well is AN EVIL when he does not know what he is doing. By saying he is evil we mean he does harm with evil intent. A God who is AN EVIL is not a God any more than one who is evil is.
God in theory is okay but living as if there is a God is illogical. So to be clear, THERE IS A LOGICAL ARGUMENT AGAINST THE WORSHIP OF GOD AND COMMITTING TO GOD. A God who invites us to committed faith is calling himself evil.
ANOTHER ANGLE
So we have succeeded in showing that a good God who is about faith and having your commitment is proven evil if he exists because he lets evil happen. But can we show that God is just incompatible with evil full stop unless he is himself evil?
Evil is that which is rejected by goodness and which rejects goodness. Evil is essentially a type of rejection. It is rejection first and rejection of good second.
If God is goodness then evil and God cannot agree.
Does that mean God can still let evil happen? Religion says that evil is not a priori incompatible with God or goodness.
That is the question.
So religion defines evil as that which is against God.
But they refuse to define it as that which contradicts the existence of God.
Thus when they say evil and God agree they are sneaking in an assumption which is that evil is never bad enough or important enough to be a disproof of God.
They argue that evil is the absence of good - in other words it is good in the wrong place and wrong time. But if evil is rejection then it is a rejection that is in the wrong time and place. That makes it about when and where it happens not about what it is. That is really a denial that evil exists. It is about mistakes not moral evil. To argue that mistakes are moral evils is an act of unjust hate. Thus in order to believe in God one has to be riddled with passive aggressive hate.
Finally
The logical argument that a really good God cannot be all-good and all-powerful and allow unjust evil or any evil is correct. It is also good. Also to argue that God and evil fit depends on calling you to redefine harm for the argument itself is insulting and incoherent and harmful. A logical argument that is evil that tries to argue God lets evil happen and remains good is self-defeating.
God is not only philosophically disproven but certainly demonstrated to be a toxic doctrine. Let us end by saying that the suggestion that evil may fit or fits a perfectly good God is intrinsically evil for God and evil are logically opposed. Why? Because evil likes to look attractive and good so you would expect it to feign how weak it is and how it has no power before God. If there is no God evil would want us to think there is so that we write it off as a cogwheel in the divine plan. It looks necessary or worthless or both.