Sorry it has taken me so long to get this out! My father's been in the hospital, and I flew down to be with him.
One thing everyone should do is think of what would make them change their beliefs. It's a tremendously important part of being human. Our first duty is to make sure we understand the world as well as possible. Without that, there's no way we could ever live up to any other duty.
I'm an atheist and a naturalist, which means I am unconvinced by the current arguments for the existence of any actual god (the atheist part) and I think that every phenomenon can eventually be completely explained by matter, energy, and the patterns they make.
What would make me believe there is a god or some sort of supernatural entity, from magic, to ghosts, to souls? In a nutshell, the balance of evidence would have to be pointing towards that conclusion. This is really the criteria that everyone uses, and everyone (I believe) would join me as an atheist and naturalist, except most people either don't have enough evidence or artificially limit the evidence they consider. It's common for people ignore the things that would lead them away from the conclusion they want. I strive not to do it myself.
Before I go into details about what would convince me of the theists' gods or the supernaturalists' spirits, ghosts and horoscopes, let me give you some of the things that wouldn't convince me.
First, personal testimony wouldn't convince me. It's not that there's anything wrong with personal testimony, for simple, everyday things. If my wife told me she had a salad for lunch today, I'd have no trouble believing her. If, however, she said that a little green person with butterfly wings -- what some theists call a "fairy" -- crawled out from under a lettuce leaf, I'd actively disbelieve her.
So, what makes the salad believable and the salad-fairy unbelievable? The rest of my experience, particularly the rest of the personal testimony I've had throughout my life. Not only would I have to ignore all of the rest of the fairy-free salads I've ever eaten, I'd have to ignore all the people who have never mentioned a fairy and told me they've never seen a fairy. I'd have to ignore the lack of books with titles like "The Lifecycle of the Salad-Fairy", I'd have to ignore the dearth of videos of showing peoples' pet fairies, the lack of pinned-fairy collections in museums. We've explored the world pretty well, and no one's mentioned anything that really match up with the "fairy" my wife claimed to see.
So what would I think it was? She might be setting up a joke, or reporting a dream or hallucination or having some sort of neurological disorder. All of those would better jibe with all of the evidence I already have then what my wife claimed to see.
Let me give another example. Imagine you're a native of an island in the South Seas, a bucolic paradise with plentiful food, crystal-blue water and a temperature that never goes below 60° F or above 80° F. You've lived your life fishing, gardening and getting drunk on palm-wine. You know that there are some other islands with people much like yourself a few day's sail away, but you've never gone there because you've never seen a need.
Into this Eden comes a shipwrecked English sailor whose name you pronounce "Dah-na-ee-ell". He is strangely dressed (in that he is dressed at all), he speaks a strange language, and has strange customs (such as staring at your women's bare breasts.) Since you have plenty eat and have nothing else to do with him, you take him in as one of your own, teaching him your language and your ways. You find he is honest, kind and hard-working, and when he becomes your sister's life-mate, you are thrilled he could become part of your family. While he has a mischievous sense of humor, you could never imagine him lying to you about anything consequential.
By far the most bizarre thing Dah-na-ee-ell tells you is that in England there is a strange rock called "Aye-suh", which has wild, magical properties. It can keep food from going bad, and makes drinks taste better, but if you hold it in your hands it can make your fingers, toes and nose fall off. It can fall from the sky in little chips or big balls, and a man can run across its surface without moving his legs if he ties knives to feet. The strangest thing of all ,this rock, this "Aye-suh" is the same things as water! Water, both fresh and salt, will turn into this rock on its own.
Now, would you, as that South-Seas native, be justified in believing Daniel? In my opinion, you shouldn't believe in Daniel's story of "Aye-suh". I've drawn a very limited circle of evidence around that little South-Sea island, and in it, all of your evidence you've seen, all of the personal experience you've heard, speak against the ice. As it stands, you have one little piece of evidence for "Aye-suh" (Daniel's stories) and lots and lots against it (everything else).
I think it's best for our beliefs to be consistent with the majority of evidence we currently have, even if that means later people with more evidence will view our beliefs as quaint and wrong. Now, of course, if that that circle of evidence expands, we should let our beliefs change. Suppose an icemaker washes up and Daniel gets it working, making enough ice to show everyone in your village. This will add a powerful piece of evidence into our circle. Once we see that this "ice" exists, we can believe it's more likely that Daniel is telling the truth about the rest of his tale.
Another thing that would have trouble convincing me -- and this might be counter-intuitive -- is personal experience, especially if that experience is one-time, fleeting and out-of-line with the rest of my circle of evidence.
Put yourself back on that island. Imagine a chunk of ice fell out of the sky in front of you, and you picked it up. You'd feel the coldness in your hand (an experience completely new to you), you'd watch as it melts away into water, and everything else. Should you now believe Daniel's claims? Perhaps, but not definitely. This is a piece of evidence, but you now that you have dreams every night that are just as real-seeming as the ice.
Wait, you might say, what's the difference between the chunk of ice that fell from the sky and one of the chunks that Daniel made from his icemaker. Nothing is difference about this ice: ice is ice. What's difference is the context of the ice. Daniel's man-made ice is reliable, repeatable and shared among all the people of the village. We could compare notes, get examples whenever the ice-maker is working, and see that it was a thing that recurred in our life, unlike dreams. All these things increase the weight of those evidence.
So, what would make me believe in some of god, then? In some sort of magic or supernatural force?
One of the most common evidences that theists and supernaturalists give to show their beliefs are true is emotional experience. If you worship, with us, you'd find that the spirit will move you and you couldn't possibly not believe.
The problem with this is that everyone says that. The Baptists say that their church services drive people into miraculous feats like speaking in tongues and fondling snakes. The Mormons feel a "burning in the bosom" when they pray on the truth of the book of Mormon. The Catholics sense the Virgin Mary and all the saints in their incense-shrouded convention centers.
It's not just Christians, either. Jews and Muslims both have their ecstatic traditions. Hindus believe Ganesha, the elephant-headed child (or, as seems more likely to me, a child-bodied elephant) clears obstacles, and Buddhists say that if only meditate as they instruct you will realize that life is an illusion. Even I, as a secularist for my entire life and a self-professed atheist for the past six years, feel things which seem to be identical to those numinous experiences when I watch or participate in art, or when I see a child or a sunset or the sea.
What if that weren't the case? What if, suddenly, every religion save one lost its savor? What if only the Jains (whose non-violence inspired Ghandi and therefore Martin Luther King) had that reverential awe and heart-stopping swell of emotion? The speaking in tongues would stop, the Buddhists would just nap, and yoga would become nothing but physical exercise. Art would become descriptive, and the Virgin Mary would become art.
If that were the case, we'd have to conclude the Jains had some sort of insight all other religions lack.
Now, I suspect that that theists -- and particularly Christians -- will say that this would somehow violate people's free will. They tend to say this whenever presented with a counterfactual that challenges their superstitions. I don't know how that makes any sense. Just because a religious claim is true doesn't mean we have to worship it. A Christian could keep worshiping her god even though it didn't give the same ecstatic cling as now, and a yogi could keep doing yoga as an exercise. In this hypothetical, I wouldn't become a Jain, because (from what I know of it) Jainism is a hypocritical and unsustainable belief system.
Think of what a wonderful, pleasant world this would be! No matter how much you learned about the world, only the Jains would have this thing they insist is proof of the truth of their beliefs. If you chose to become a Jain, you would feel nothing like anything you ever had before. You'd have a feeling so deep, so abiding, that people who insist on worshiping in other houses could never even describe. You'd be a sighted person on a planet of the blind.
Now, if you're a theist, tell me: why doesn't your god run things this way? He (or she or it) wouldn't be undermining anyone's free will, and would be paving the way for people to make their way to him (or her or it).