Here is a way of looking at the inductive problem from evil.
Start with this. Let E be the bare claim that there is at least one evil. Let T be theism and N be naturalism, and suppose those are the only two relevant options. Then, I think one can argue that P(E|T)>P(E|N). Why?
- The existence of teleology or proper function is certain on T (at least God will have a proper function, viz., of being his perfect self). Plausibly, evil requires teleology or proper function (evil is a way of something's going wrong) and teleology and proper function are unlikely on N.
- Alternately: evil requires consciousness, and consciousness is much more likely on T than on N.
- It is likely that God would create a large number of significantly free persons, and it is likely, given a large number of significantly free persons, that at least one would do wrong in some way. On the other hand, it is not all that likely that there would be a morally responsible being if N were to be the case.
- Many significant goods, like courage, forgiveness and perseverance, require at least some evil. Thus it is unlikely that God would strive to create a world where evils would be certain not to occur.
- There is some plausibility in significant contrast stories on which experiencing a minor evil can enhance one's enjoyment of goods. On T, this renders evil more likely, while N doesn't care about us enjoying things.
This is how the inductive problem from evil should be seen. The mere fact of evil is evidence for theism, but we need to ask whether this evidence for theism is strengthened, left unchanged, weakened or even flipped the other way around when we look at the details of the sorts of evils we observe. For each of these four options is in general possible when we start with some general piece of evidence and then look at the specifics.
Nonetheless, it will make a difference to how we think of the specifics if we start with the idea that the general fact of evil supports theism, and we ask whether this support remains or flips when we look at the specifics.