My position is that propositions become our evidence via the deliverances of basic faculties such as the senses, memory, introspection, and rational insight. Without the latter, we have little or no way of extending our basic knowledge beyond the testimony of the senses. The principles of inference according to which we extend basic knowledge are justified because of the testimony of rational insight. [Let's bracket for now concerns about Carroll's Paradox, because I don't think that will affect my point.] Rational insight can be mistaken, as in the apparent self-evidence of the naïve axiom of comprehension, but its reliability is entailed by our general reliability.
Rational insight sometimes gives testimony to simple facts--like the transitive property of equality--but sometimes to more complex items like the solutions to mathematical or metaphysical problems. You might seem to "see" that there could be a proof of a theorem, a solution to an equation via number theory, or a way to use probability to explicate justification.
Many readers will have had the experience of many failed attempts to make such apparent insights precise in a system of logic before apparently reaching success. At some point, of course, repeated failure is enough counter-evidence to undercut and defeat the evidence provided by the testimony of reason (or the deliverances of reason will be more clear for the negation, as in the case with Russell's Paradox). When it does so is vague, but I think at times, given a plausible view of our resources, it would take a lot of failure to do so.
In fact, I think that sometimes repeated failure is evidence for the insight when it is repeated failure by multiple people. Think of the history of failure to prove Fermat's last theorem. Personally, I never doubted the theorem for a second and I doubt I am alone in believing that the repeated failure to provide a proof did not provide much if any evidence that it was false. Or consider what a history to prove Goldbach's conjecture would look like (I haven't looked to see if there is an actual history of attempts to do so). The very fact that so many people have the insight that it is true is what is guiding all these (sadly failed) attempts, and the (partial) independence of the testimony can be surprisingly strong evidence when modeled probabilistically. And it helps when there is considerable conceptual similarity among the attempts, for the insights are often of the form "considerations pertaining to X support Y" (and we just can't get the bridge in formal logic yet).
Now think of the history of attempts to prove God's existence by, say, the Ontological Argument (OA). I think there is a sound OA, but suppose you don't. There are OA's in Augustine, Anselm, Descartes, Leibniz, Hartshorne, and Plantinga et al. And even if you think they are none of them so much as valid, I think their (hypothesized) failure would not mean that one lacked evidence for the existence of a sound OA. I think the fact that partially independent sources (not to mention what sources they are in terms of quality and otherwise disparity of conceptual framework!) have the testimony of reason that there is such an argument to be had is evidence that it is so, and I think that it is greater than any contrary evidence which might be provided by the failure to make precise that insight in the language of logic (much as that is to be desired). I think this is even more so with at least two versions of the cosmological argument. The question, for those of us seem to have the insight that considerations of causation, infinity, and contingency point to a supreme being isn't so much whether the insight is or is not true--it maintains its luster of truth (too bad "truthiness" is taken!)--but just how to show that it is true. But knowing doesn't require showing (much as that is to be desired).
I'm inclined to think that many ordinary beliefs by ordinary believers depend on this kind of scenario and that meditation on the history of math and science bear out this epistemological assessment via parallel examples. I'm also inclined to think that though there may be some atheological parallels, the theist has the advantage.
