How though do we know that the Bible is God's Word? Gerhard isn't as excited by answers like those given to Chemnitz in Examination of the Council of Trent. In that work, Chemnitz mainly makes the argument in the first volume, that the Scriptures are reliable kerygma because the historical evidence says that they are Apostolic. Similarly, the Apostles were vindicated by the fact that they did miracles. This is actually an old stock argument from the Middle Ages (Aquinas in pt. 1 of the Summa uses it as well), with a new Lutheran twist. The Lutheran twist is that only the Biblical material has been vindicated by these historical proof, not the unwritten traditio (which he lambasts as the only form of invalid traditio, out of 8 he outlines).
Of course, from the perspective of a Protestant living in the wake of Bellarmine (like Gerhard was) one can see why this argument might be less attractive. Bellarmine had emphasized that the authority of Scripture was not a good one because at the end of the day it was dependent on Church attestation and therefore authority. In the early 17th century, it is easy to see why this historical line of reason declined (it totally disappear in a thinker like David Hollaz at the end of the 17th century- He also, in keeping with this, doesn't even mention the distinction between homologoumena and antilogoumena!). Scriptural authority had to be independent of Church authority, even mere historical attestation, as much as possible.
While not rejecting historical arguments (as anyone who has read the Confessio Catholica knows), Gerhard is more interested in the argument from the inner testimony of the Spirit. Since as Paul writes, it is the Spirit that testifies with our spirit that we are the children of God (Rom. 8:16) and he does so through the preaching of the Word (Gal. 3:2), clearly the Spirit is the one who testifies that the Word we are hearing is God's Word- because what it says is true and from God. This isn't a new argument of course. Both Calvin and Luther had both made this argument and it was also present in a number Patristic authors as well.
For Gerhard, then, the principium of Scripture is to theology the way that numbers are to math. In fact, he makes this analogy. Just as to do math at all you would need numbers, you need the Bible to do theology. This of course brings us back to the question of why Scripture is authoritative. It is, he says, for the same reason that numbers are- theology as an enterprise just doesn't happen without it. Neither can you prove it in an ordinary sense, in the same way that it is absurd to say to someone "Prove 5." 5 is a self-evident principle. You can't ground it in anything because it is the principle. Your knowledge of the world intuitively and functionally accepts "5." In the same way, Gerhard says, that we possess an intrinsic knowledge of "5" through the Spirit convicting us through his inner-testimony, we too who are Christians have an intrinsic knowledge that the Bible is the Word written into us.
So, does this work? I would say yes (based on the exegetical argument), but I would also make some other observations.
First, the Word is not quite self-authenticating thing the way that math is, since it comes from something external to ourselves (the Word working through the power of the Spirit), rather than something built into us. Secondly, it's not universal the way math is. It's particular to the elect.
This though, I think, is the basis for it being a proper principium and self-authenticating. In other words, we're certain that math is true because everyone knows it to be true. It's built into us. Conversely, we know that the Word is true because everyone doesn't know true. If it has been revealed by God, presumably it's something alien. Otherwise, why reveal it? It's not something you would dream up. If it were, then you might begin to doubt that it really came from God and wasn't just a projection of our inner consciousness (the old Feuerbach critique of Schleiermacher).
But why isn't it something that we wouldn't just make up? Precisely because in the opposition to math, it's counter-intuitive. In other words, it tells us that we are completely unrighteous because we have violated God's law. Being that in all our experience of the world, the law is there to tell us what to do, and therefore assumes we can do it (a point made both by Pelagius and Kant!), this is counterintuitive. Why tell someone that they are sinner if there's no hope for them getting better? Also, why would we believe a something makes us think so lowly of ourselves? Humans always want to think the best of themselves, yet we are told that we are the worst we can be and for some reason we believe it.
Secondly, it tells us the gospel, which testifies that we are completely forgiven by God's grace alone. This again doesn't make sense in according to our analogy of experience. Gods as a rule accept sacrifices, they do not sacrifice themselves to us. This works well with how religion normally functions. As a form of social manipulation, religion helps keep people good and the priests fat by telling people the good works they can do and sacrifices they can perform. In fact, all the gospel does is give free forgiveness and therefore freedom. It doesn't prescribe anything. Also, it takes away all our power to manipulate the gods/God, because it says "hey, it's already done. You can't buy me off."
For this reason, Gerhard makes the correct analogy. Nevertheless, the principiums of theology function exactly in the opposite way of other principiums. The principium of theology works precisely because it is counter-intuitive, rather than intuitive.