I'm going to comment on part of this:
Unlike Van Tilian presuppositionalism, classical presuppositionalism will not argue, “God exists, therefore God exists.” It will not argue, “The Bible is the Word of God, therefore the Bible is the Word of God.”
Can Beisner quote examples of Van Til, Frame, Bahnsen, David Byron, James Anderson et al. who actually argue in that fashion, or is that Beisner's caricature?
Instead, classical presuppositionalism affirms that logic, or reason (terms it considers synonymous), is simply the structure of God’s thought…
Are reason and logic synonymous? Consider modus ponens:
If P, then Q. P. Therefore, Q.
Is that synonymous with reason? Isn't modus ponens just an abstract rule of logical inference? It's purely general. It has no specific factual content. That, of course, is part of what makes it useful. You can plug so many different things into the formula.
But isn't there more to divine reason than fact-free laws of logic? For instance, doesn't God know all truths of fact? Particular facts about the actual world–as well as possible worlds?
1. The Bible claims to be God-breathed.
2. All explanations of the claim other than its truth are untenable.
3. All attempts to refute the claim by pointing to specific errors in the Bible fail.
4. Therefore we are justified in believing that the Bible is true and God-breathed.
Consider the argument piece by piece. First, the Bible claims to be God-breathed. To note this is not to argue in a circle; it is merely to set aside the hypothetical objection that we are claiming for the Bible what it does not claim for itself. It would after all be rather gratuitous to claim that the Bible was the Word of God if it did not claim to be. “There is no reason for making assertions beyond those that can validly be inferred from the statements of the Bible,” Clark writes. “. . . What the Bible claims about itself is an essential part of the argument. The Christian is well within the boundaries of logic to insist that the first reason for believing in the inspiration of the Bible is that it makes this claim.” Clark cites, among other passages that (explicitly or implicitly) make this claim, 2 Timothy 3:16, John 10:35, 2 Peter 1:20, 21, Romans 3:2, Matthew 11:9-15, Romans 16:25-27, and Ephesians 3:4-5.
i) An obvious gap in Beisner's argument is that books like Matthew, Luke, and Acts don't claim to be inspired.
ii) Perhaps Beisner would try to close the gap by claiming that they are covered by categorical statements like 2 Tim 3:16. That, however, only pushes the question back a step: if Matthew is Scripture, then Matthew is inspired. But how does Beisner determine that Matthew is Scripture? How does he determine that 2 Tim 3:16 includes Matthew in the scope of its claim?
Provided the Protestant canon of Scripture, suppose Beisner's argument goes through. But how, given Clarkian epistemology, does he determine the canon in the first place? Consider Clarkian demotion of historical evidence and probabilistic reasoning. In making a case for the canon, how does Beisner avoid the methodology of evidentialism?
Second, all explanations of the claim other than its truth are untenable. Consider three possible alternative explanations. One is that the claim is only occasional and accidental and therefore should not be taken seriously. But a careful inspection of the Biblical data, e.g., as done by Louis Gaussen in his Theopneustia (published in translation as The Divine Inspiration of the Bible) or by Benjamin B. Warfield in The Inspiration and Authority of the Bible, demonstrates that the claim is pervasive and crucial to much of the rest of the program of Scripture. It is therefore not accidental and cannot be trivialized or ignored. A second is that the claim is one among many by writers whose other claims provide good grounds for skepticism about their credibility, and therefore the claim lacks a priori credibility. Yet a careful examination of the writings indicates the opposite: that the writers were highly credible on other matters and made this claim in complete awareness of what they were saying, and therefore that the claim’s falsehood is unlikely a priori. A third is that though some other Bible characters might have made the claim, either Jesus did not make it or, if He did, He made it only in accommodation to the prevailing views of his contemporaries, and since Jesus is the most important character in the Bible, His failure to make the claim renders the claim unlikely. But again, careful inspection of the data indicate that Jesus did make the claim, that He did not do it merely as an accommodation to His contemporaries’ prevailing views (Indeed, He was quite in the habit of contradicting prevailing views that He considered wrong!), and that He made the claim in full self-awareness. Therefore, if the claim is false, it becomes an evidence against Jesus’ credibility. Yet Jesus’ credibility is otherwise impeccable. Therefore Jesus’ credibility gives His claim a priori credibility. Perhaps there are other alternative explanations that need examination, and in the appropriate context that could be done. But for the sake of illustrating the method, the consideration of these three is sufficient. If there are four and only four possible explanations of a phenomenon (in this case a claim), and if three of them can be shown untenable, then it follows that the fourth is to be affirmed. Thus, it follows from the failure of alternative explanations of the Bible’s claim to be the Word of God that the Bible is in fact what it claims to be: the Word of God.
Notice the repeated appeal to "credibility." But how is that essentially different from the probabilistic methodology of classical, evidentialist, and cumulative-case apologetics?
Third, all attempts to refute the claim by pointing to specific errors in the Bible fail. A study of individual examples of alleged contradictions in the Bible, such as John W. Haley’s Alleged Discrepancies of the Bible, demonstrates that none of the allegations proves true. Similarly, a study of individual examples of alleged historical inaccuracies in the Bible, such as we find in Gleason Archer’s Encyclopedia of Bible Difficulties, demonstrates that none of those allegations proves true.
Once again, doesn't that boil down to probabilistic reasoning? Doesn't that appeal to things like archeological corroboration? How does that distinguish his position from classical, evidentialist, and cumulative-case apologetics?
Fourth, therefore we are justified in believing that the Bible is true and God-breathed. This follows from the first three premises, and the argument is noncircular.
Even if that's the case, isn't that basically inference-to-the-best explanation? But doesn't that fall well below the exacting standards of Clarikian epistemology?