The key to understand here is that Van Til does not accept that there is a neutral reason “out there” by which Christians and non-Christians can arbitrate the truth of Christianity; his point is that there is no neutral ground and that one’s presuppositions will determine one’s end point. Again, he writes, “this [apologetic method] implies a refusal to grant that any area or aspect of reality, any fact or any law of nature or of history, can be correctly interpreted except it be seen in the light of the main doctrines of Christianity” (Christian Apologetics, 124).

However, Van Til takes it even further and argues that one must presuppose the truth of Christianity in order to make sense of reality: ” What is the content of this presupposition, then? It is this: “I take what the Bible says about God and his relation to the universe as unquestionably true on its own authority” (The Defense of the Faith, 253); again, “The Bible is thought of as authoritative on everything of which it speaks. Moreover, it speaks of everything” (Christian Apologetics, 19). Thus, Van Til’s apologetic does not make Christianity the conclusion of an argument; rather, Christianity is the starting presupposition.

The presuppositional approach here cannot be stressed enough. For Van Til, one simply cannot grant to the non-Christian any epistemic point. “We cannot avoid coming to a clear-cut decision with respect to the question as to whose knowledge, man’s or God’s, shall be made the standard of the other. …[O]ne must be determinative and the other subordinate” (The Defense of the Faith 62-63).

What I'd like to know is why this is the only option. What actually rules out the possibility of "neutral facts" in the sense of facts that could be accepted by both Christians and non-Christians?

To appreciate this question, consider a couple epistemic models. Suppose on the one hand that the claims Van Til seems to be making are correct: there are two and only two ways of reasoning, "man's" and "God's". If Christianity is true, and if these are distinct (and distinguished by their results), then plainly God's way of reasoning will get at the truth and at Christianity, while adopting the other will not. Fair enough.

But on another model, we might acknowledge that there could be overlap between man's and God's ways of reasoning, even if they are distinct and distinguished by their results. They might share some premises or rules of inferences or ways of balancing evidence or what have you. As long as they differ in other ways, they could still yield distinct conclusions about Christianity. But that possibility would contradict the view that there could be no overlap, and would leave open the possibility that one could argue from that common ground to theism.

The summary to which you have linked does not seem to contain an argument that this sort of overlap is impossible. I don't see why it should be entailed by Christianity. Indeed, it's not clear what purchase the 'transcendental argument' mentioned here would have unless someone could appreciate the epistemic weight of "morality, science, history, and rationality" prior to accepting Christianity.

A third epistemic model might be the following: there's one standard of rationality, shared by all humans, but some humans, due to error or vice, fail to live up to it and therefore don't arrive at theism. What rules that possibility out?

My other sense, as I read this, is that some of the claims made are fairly mundane but are interpreted as "presuppositionalist arguments." It strikes me as somewhat tendentious, for instance, to call the transcendentalist argument a "presuppositionalist argument." It rather just seems to be claiming that non-Christians cannot "save the phenomena." Another example:

The key to understand here is that Van Til does not accept that there is a neutral reason “out there” by which Christians and non-Christians can arbitrate the truth of Christianity; his point is that there is no neutral ground and that one’s presuppositions will determine one’s end point.

Some of these claims are ambiguous between stronger and weaker readings. Of course there is one sense in which one's presuppositions determine one's end point, but perhaps some of one's presuppositions are correct and others are wrong, and it's possible to see how some are wrong.

It's likewise a truism that one can concede too much to the non-Christian, and that is a risk in apologetics.

My sense is that the 'theory' about not being able to concede anything at all to non-theists is pious (in the derogatory sense) confectionary on top of the acknowledgement of rather mundane points.

I admit that I haven't read anything by Van Til, but I had some friends at one point that were into this kind of thing.  I took the idea to be rather similar to MacIntyre's in Whose Justice? Which Rationality?: simply that there is no getting outside intellectual tradition in evaluating competing ultimate claims to truth and ways of looking at the world, but this does not mean that we have no means of evaluating competing traditions because we find ourselves already engaged in dialogue.  Said another way, there is no way to evaluate worldviews in a worldview-neutral way, but we can still see that some worldviews make better sense of the world than others.  (Like Greg, I think that so far we are at the level of truism, albeit a relatively interesting one.)

When I have talked to these friends about the incoherence point you mentioned, the answer I got was that Van Til is talking about incoherence within the space of debate and dialogue.  The very act of engaging in conversation about the truth or falsehood of Christianity makes some positions dialalogically untenable (e.g. that there is no such thing as truth).  I take the idea to be similar to the Socratic elenchus.  There is, of course, nothing Socrates can really do to make Euthyphro stay and finish the conversation, much less agree with him, but he can push on the awkwardness of Euthyphro saying one thing and then saying the opposite.  The characteristic step of Van Til and his followers is to hold that ultimately, if we keep on pressing the conversation, Christianity is the only conversationally tenable ground for the conversation itself and that either we have been tacitly presupposing the main lines of Christianity all along or we have somehow been acting in bad faith by the very act of having the conversation.

I take the point of saying "presupposition," instead of just "supposition" or "belief," to be an emphasis on what Michael Polanyi calls the "tacit dimension," that is the thick background of unacknowledged assumptions behind every explicitly stated proposition or action.  I think the idea is that we are supposed to acknowledge this background behind the very act of engaging in a conversation about apologetics.  If we do this, we are supposed to see that any anti-Christian claims are at least in tension with this background while Christianity alone can make sense of it.

I'm not a supporter of this approach, just thought I would pitch in my two cents at trying to understand this way of thinking since I think it holds a lot of traction in certain conservative reformed circles (not that I'm reformed either).

So I started to formulate a response and halfway through the first sentence convinced myself that I have no idea either.

Speaking for myself, I think that all dialogue requires a commitment to the link between λόγος in us (both in thought and in speech) and λόγος in the world.  Ultimately, this depends on the Λόγος as the ground of these many λόγοι.  As the ground of being, I think we can show that this Λόγος is God yet is also the Λόγος of God.  Only Trinitarianism (and a fortiori only Christianity) makes sense of this.

I'm not sure where I stand on Greg's claim that coercive evidence would preclude faith since I tend to think of faith in personalist terms, so I'm not sure what I think the force of this line of thought is.  Nevertheless, this is essentially the line that I think runs from "We're having a conversation right now" to "So therefore Jesus Christ is Lord."

Jeremy Taylor wrote:

But what would stop the Scientologist or Mormom making similar arguments? This seems a problem for any basically fideist position. If there is an appeal to the greater understanding of the world orthodox Christianity can offer, then doesn't that assume some shared standards for judgement are possible?

^ This.