It generally takes this form:
Skeptic: To the extent that we can apply reasoning, I can prove X, given Y. We all agree that Y is the case, so X is the case.How does this sort of defense work?
Believer: I agree that Y is the case, but I deny X.
Skeptic: You object to my proof?
Believer: No. I deny your ability to apply reason to the problem.
Skeptic: Are you saying the world is illogical?
Believer: No. The world is logical.
Skeptic: You are agnostic about X?
Believer: No. I deny X.
Skeptic: Well, if you deny our ability to apply reason to the problem, how do you obtain a reasoned inference to your disbelief in X?
Believer: It's an inference to the best explanation.
Skeptic: Huh? How can you have an inference without reason?
Believer: I have lots of reasons to disbelieve X. But I still deny your ability to apply reason to the problem.
The Alleged Limitations of Models
People are used to thinking that symbolic reasoning is very limited, and cannot be applied in many circumstances. Many times in debates, theists have argued that human thinking and behavior cannot be modeled statistically. This is false. The actual fact is that human behavior cannot be predicted with total precision (at least not yet), but that's not the same thing as statistical modeling. For example, if it were Freaky Friday, and your wife swapped bodies with your daughter, you would likely see the difference because their behaviors would deviate from the statistical norm. Indeed, without statistical models, there would be no such thing as personality.
Unwillingness to Question Assumptions
The theist's denial of our ability to model a thing is common whenever the thing is allegedly magical. According to most theists, humans are magical because they have magical free will, magical reasoning ability and magical moral sensitivity. It's fair to say that the theist's intuition is that these things are magical. That is, it is a natural, unjustified belief that free will is magical.
Yet, philosophy is all about questioning our beliefs. If a belief is not formally justified, we ought to question it. In the case of free will arguments, theists argue that our experience free will requires that the world not be deterministic. This is unsupportable because, if the world were deterministic, no one would notice the difference.
When I am able to make a very persuasive Bayesian probabilistic case for my position, my argument is rejected because the odds have been tipped in circular fashion. This applies across their entire world view.
There are generally three categories of evidence that are cited by believers: historical, personal, and philosophical. None of these alone has persuasive force, and some theists admit that you have to examine the big picture. Unfortunately, this big picture examination consists of circular reasoning.
It goes like this. Examine the historical evidence. The evidence, say, for the Resurrection of Christ, is utterly abysmal. Even if we thought the odds of the Bible being a fabrication were a million-to-one against, we would still rationally have to believe that the odds of the Resurrection were at least a thousand-to-one against. "Ah," say the theists, "that's only if you ignore the philosophical and personal evidences! However, I don't have time to go into them all in this discussion of historical evidences."
Sounds potentially reasonable. Yet, when we discuss the personal evidences for miracles, and I point out that it is irrational to believe that a particular event was an answered prayer, the theist refers me to the philosophical and historical evidences (which they can't go into at that moment).
And, of course, when we discuss the philosophical arguments, we are told that symbolic rational inquiry is useless, just as it is when applied to the historical and personal lines of evidence.
The theist world view is like an M.C. Escher drawing. Each line of evidence looks as if it's supported by the other two lines, but in the grand scheme, none of the lines can support themselves.
Is there still a refuge left for the theist? One might argue that every world view relies on some unprovable statements. I think this is so. My unprovable truths are the axioms of rationality. I'm happy to state them, and happy to admit that I can't rationally prove the laws I use to reach rational conclusions.
However, the arguments against the theist's lines of evidence are not just neutralizing. They're destructive. For example, it's not rational to believe that it is likely that Christ was resurrected, or that the sunny weather is God's answer to a prayer, unless you're willing to believe lots of other improbable things are actually the case.
While the theist can't make a case just from the axioms of rationality, the theist could add in some new axioms. One axiom could be that, say, Christ was resurrected, even if, by the evidences, that's improbable. Another would be that, when I feel like a prayer is answered, it is answered. If these axioms are added to the theist's world view, the evidences will support their conclusion.
I think it's clear that this is a losing proposition. I can prove any conspiracy theory I want by resorting to additional axioms that have no supporting evidence. For example, a theory that JFK was shot by Martians can always be rescued by proposing the axiom that Martians exist, and the axiom that they have it in for any leader who meaningfully supports the space program. But this is absurd.
Sure we can always get to a conclusion by proposing axioms that make the conclusion necessary. We don't do it because, if we did, we would suck all the utility out of rational inference. The utility of rationality isn't merely in its ability to portray a consistent picture. It's power is in its ability to tell us which picture is more probable.