Some religions will be centered on points which are undecidable. They cannot be undermined by logic because logic cannot prove or disprove their constituent beliefs.
Others may center on points which are decidable. These religions are either the strongest ones, if every point is provable, or the weakest ones, if some of their central points are provably false. Feel free to disprove points of a person's religion. As various people have said above, fill in the void you leave. I would not, though, disprove each point at once. Let them shift slowly.