I'm going to post once and you can feel free to consider or disregard my thoughts. I won't be checking back in on my post because I have neither the time nor inclination for debate tonight.
If the events of the Acts of the Apostles occurred today, I think that they could, in some measure, be used as a proof. Chronic illness healed, death undone, prison doors opened, and so on. (If you haven't the vaguest idea of what I'm talking about, feel free to read the first portions of Acts.) Naturally, this doesn't make up a good argument today because there's no way of fully judging the accuracy or truthfulness of the written accounts. Nor, even so, would that allow scientific experimentation on the subjects.
In like manner, one of my own personal arguments for the existance of God, and I speak now of an argument I use with myself, with no intention of convincing others, is that I have seen and encountered some similar things. This isn't much of an argument for others because, as with the written account, it is a testimonial that cannot be scientifically examined in a way to satisfy the scientific method. However, when you have had the chance to see, touch, examine and apply an informal scientific method, it becomes convincing for the single observer. My own arguments for God, when I argue within myself, include the evidence that I have seen by way of cause and effect that mimics the stories in Acts, and in the gospels.
I don't think it's a particularly good way of collecting information to ask a lot of unschooled people to provide scientific evidence; a forum of scientists would be the better place for that. As for logical evidence, your own link dooms this discussion from the beginning. "... based on the truth of a set of assertions called premises." The particular groups that involve themselves in this discussion will never see logic in each others' statements because they will not, and perhaps cannot, agree on a shared set of premises.