I can understand naturalism and its process of explanation. However, many people are unhappy with it and seek supernatural explanations of events. But what is a supernatural explanation? When should one believe in one? Does it really explain anything?
Consider the case of a person with late-stage cancer. He is prayed over and then he recovers and his cancer goes into remission. Is that a supernatural or a natural healing? What does supernatural healing mean?
First one could define natural healing. This could be the body's immune system managing to isolate and kill the cancer cells, or a medical procedure where an instrument is used to remove the tumor or some other physically describable process. A supernatural healing, by definition, cannot be only this. During some stage of the healing there must be something that happens that does not follow the normal laws of physics. At some instant some molecules realign in a way that they never could under normal conditions requiring a force beyond anything in the known universe. For this reason we call it supernatural.
While many people believe in supernatural explanations, I presume most would agree that almost all events (>99.999%) are natural. That is, given an event, our presumption should be that it is natural unless there is strong reason to suspect it is not. So what is a good reason to suspect a supernatural explanation? Back to the late-state cancer example. Most people with late-stage cancer die because of it. Occasionally some people's bodies unexpectedly manage to recover for some natural cause like the immune system winning. Now many people with late stage cancer are earnestly prayed for, but despite this most of them die (otherwise it wouldn't be such a deadly killer). Sometimes someone with late stage cancer is prayed for and then he recovers. Being prayed for and then being healed is a very unlikely event and so one might say it is evidence of a supernatural healing. But this ignores all the negative cases of people being prayed for and not being healed. If there are occasional unexpected natural recoveries, then one would expect that sometimes these would occur after someone is prayed for, especially if they are prayed for for a long time. So positive answers to prayers aren't good reason to believe in supernatural explanations, if there are many negative answers to prayers that go unnoticed.
Perhaps evidence for the supernatural is when an individual has great healing power. If so then we ought to take Benny Hinn seriously; at least his followers would point to his many great healing miracles (see his web page for examples). Others would be more sceptical suspecting that there are many people who aren't healed and wonder if the healings don't have natural explanations despite the claims. Well, then one could point to Jesus; he surely had many great miracles. But here we only have the testimony of his followers and they are unlikely to record failed miracle attempts, and in the one case where this is hinted at, (Mark 6:5-6 "He could not do any miracles there, except lay his hands on a few sick people and heal them. And he was amazed at their lack of faith."), the explanation given was lack of faith. That is a convenient explanation for any healer when he fails to heal. Unfortunately we don't have any measure for what percent of people were healed or how effective the healings were. Without those negative data we can't declare the healings that worked as evidence for the supernatural.
Then what about miracles that can't have natural explanations. Many religions have examples of these. But the high likelihood of events having natural explanations should cause one to have great suspicion of these events and consider possible natural explanations like the stories were embelished by the followers or recorders. I think David Hume's argument is that the latter is much more probable than the events actually happening as told.
Nevertheless, say there is a well established event that can't have a natural cause and so we say its cause was supernatural. What are we gaining by calling it supernatural? Aren't we simply saying that we can't understand how the event could have happened given our model of the natural universe? Of course people like to anthropomorphize the spiritual world; fill it with human-like demons and angels, gods and goddesses, or a human-like God. People preach all kinds of things about the supernatural, but isn't this far more likely to be wild speculation than true knowledge. How could anyone know what the supernatural actually contains? Perhaps he feels it, senses it, has a vision of it; in my opinion all ways of "knowing" that are very suseptible to self delusion and fulfilling what one wants to believe. No matter what someone's character, it seems pretty easy to be deluded about something one does not have a direct method of sensing. So I think we are stuck being ignorant about the supernatural.
Here are my conclusions. First almost all events are natural, and so this should be our default assumption unless there are strong reasons to believe otherwise. But it is very hard specify reasons to believe something is supernatural. An event being improbable is insufficient to rule out natural explanations, as many natural events are improbable. Now even if one can show an event is supernatural, all that one is saying is that one cannot understand the event in terms of natural causes, and this is not really an explanation.
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