How may we demonstrate that the Bible truly is a witness to the Word of God? By the inner testimony of the Holy Spirit alone! Karl Barth affirms this famous phrase from John Calvin because the only authority that may demonstrate that the Bible is of divine authority is God himself, and any other proof whether miracles, church history, inerrancy or the like are human judgments that are not sufficient grounds to demonstrate anything is divinely revealed and these "testimonial arguments" are luxuries for those weak in faith but may not be used as a secondary ground to support the Inspiration and Authority of the Bible (contra B.B. Warfield, et al.) David F. Strauss called the inspiration of Scriptures "the Achilles' Heel of the Protestant system", and Karl Barth proudly admits that it is so, because there is no other ground for the authority of the Holy Scriptures apart from this inner testimony of the Holy Spirit.
In this post, I will explain why Barth says that the authority of the Bible may only be established based on the inner testimony of the Spirit, by using selections from the Church Dogmatics: The Word of God, Vol. I/2 (§19 in loc.)
The authority of the Bible is based on a logical circle
How may we know that the Bible is a true witness to the Word of God? Karl Barth's answer is because the Bible says so! Isn't this a circular argument? It absolutely is! And, this logical circle is necessarily so. Only God may guarantee that God has spoke in the Bible, and therefore it is a divine judgment and not a human judgment that determines that the Bible is a true witness to the Word of God. Therefore it is vanity of vanities (Eccl 1:2) to endeavor to prove the bible by the means of Inerrancy, rational cohesion, miracles, church history or any other human judgment because it is a divine judgment alone that determines that the bible is a witness to the Word of God. Barth supports this position with an appeal to the first article of H. Bullinger's 1562 edition of the Second Helvetic Confession in the following quote:
The Bible must be known as the Word of God if it is to be known as the Word of God. The doctrine of Holy Scripture in the Evangelical Church is that this logical circle is the circle of self-asserting, self-attesting truth into which it is equally impossible to enter as it is to emerge from it: the circle of our freedom which as such is also the circle of our capacity. When the Evangelical Churches of the Reformation and later were asked by their Roman adversaries how the divine authority of Scripture could be known and believed by men without being guaranteed by the authority of the Church, the Evangelical theologians gave the hard but only possible answer that the authority of Scripture was grounded only in itself and not in the judgement of men. 
Barth provides the following quotes as proof:
. . . the canonical Scriptures . . . have sufficient authority from themselves, not from men. . . . (Conf., helv. post., 1562, Art. 1). We might just as well ask where we can base the distinction of light from darkness, of white from black, of sweet from sour (Calvin, Institi. I, 7, 2). The question of whether the Scriptures or sacred books are the Word of God is unworthy of a Christian man. . . . (J. Wolleb, Chr. theol. comp., 1626, praecog. 7). 
There is no 'secondary support' for the Word of God
The most rest assuring argument for the authority of the Holy Scriptures has always been the existence of the Church, however, Barth rejects this argument upfront by turning it upside down. It is not the Church that establishes the Bible but it is the Word of God that establishes the Church.
The Church does not have to accredit it, but again and again it has to be accredited by it. And all that we may adduce on other grounds for the authority of Scripture does not underlie this one ground and its divinity, but at best can be sustained only on the presupposition of this one ground and as pointing to it. 
Barth rejects all human arguments from below, because all of these testimonial arguments are flawed for the same reason why all so-called Christian Apologetics are wrong, and that is because divine authority is divinely established alone, and is never established by any human argument, whether rational or historical or what-so-ever:
The 16th century was well acquainted with and even accepted--just as it accepted the authority of the Church under Holy Scripture—an apologetic which came down from the early Church and the Middle Ages, subordinate to but illustrating this one ground: testimonial arguments, human considerations by which it was thought that the divinity of Scripture could later be more or less clearly brought out. Attention was usually drawn to the antiquity of the Bible, its miracles and prophecies, its decisive and victorious role in the Church history. 
Calvin's Inner Testimony of the Spirit
John Calvin says that the Bible may only be established by the "inner testimony of the Holy Spirit" and Barth says amen! Calvin explains that the Inner Testimony of the Holy Spirit means that when the Bible is read, the Holy Spirit enlightens the reader with the knowledge that these Scriptures are from the very mouth of God (Institutes I, 7, 5). Calvin's phrase is similar to Barth's phrase that the Bible becomes the Word of God. So it is no surprise that Barth affirms and adopts Calvin's doctrine of the Inner Testimony of the Holy Spirit.
Barth also agreed with Calvin that there are no secondary grounds for the authority of the Bible apart from the inner testimony of the Spirit. Calvin denied that the authority of the Bible is established by any human argument or judgment, but he did permitted these secondary grounds that he called "testimonial arguments" to be used as luxuries by people who were weak in faith. Barth defend's Calvin allowance for testimonial arguments however Barth said this had a disastrous consequence in Calvin's successors whom elevated these testimonial arguments to the primary ground for establishing the authority of the Bible—even above the testimony of the Spirit (c.f. CD I/2 in loc.)
Calvin thought it necessary to devote a whole chapter of the Institutes to these considerations as they throw light on the existence of the Bible (I, 8: Proofs in as far as human reason can support). But he himself calls them secondary support for our weakness and warns us in every possible way against thinking that we can regard and apply them as the grounds of faith: they act stupidly, who want to prove to unbelievers that Scripture is the Word of God, because it cannot be known except by faith (I, 8, 13). The verdict that Scripture is the Word of God is not a human but a divine judgement, and only as such can it be adopted and believed by us (Institutes I, 7, 5). . . .
Unfortunately, Calvin found many later imitators in the enumeration and development of these secondary grounds, but not in his definitely expressed perception of the abysmal difference of these grounds from the one primary and real ground, not in his awareness of the superiority and self-sufficiency of that one ground. The internal witness of the Holy Spirit, on which alone he and the Reformation as a whole based faith in the Bible as the Word of God, at a later date gradually but irresistibly became one ground with others, and the other grounds gained an interest and acquired an importance as though they were, after all, autonomous. 
D.F Strauss said this is the Achilles' Heel of the Bible
D. F. Strauss is famous for saying that the doctrine of the Inspiration of the Scriptures is the Achilles' heel of the Protestant system. Barth agrees with Strauss and affirms this conclusion is correct by saying that God alone may testify that the Bible is a true witness to the Word of God, and there is no other human argument that may be advanced to establish it as so. Barth's affirmation of Strauss' phrase is surprising because so many theologians have tried to answer Strauss by establishing a secondary ground for the authority of the Bible rather than embracing the logical circle of its authority. Barth says that Strauss is right to see the inspiration of the Scriptures as the weakest point, but it is precisely at this weakest point where we find its indestructible strength (c.f 2 Cor 12:9-11). (For more on D.F. Strauss, see here and here.)
D. F. Strauss was right to criticize this rule: "Who can now attest the divinity of this witness? Either itself again, which is nobody: or a something, perhaps a feeling or thought in the human spirit—this is the Achilles' heel of the Protestant system"(Die chr. Glaubenslehre, vol. 1, 1840, 136). Indeed, who does not attest the divinity of this witness? What Strauss failed to see is that there is no Protestant "system," but that the Protestant Church and Protestant doctrine has necessarily and gladly to leave his question unanswered, because at its weakest point, where it can only acknowledge and confess, it has all its indestructible strength. 
Karl Barth faces the devil, and agrees with John Calvin, that we have no other recourse to defend the authority of the Holy Scriptures by the Inner Testimony of the Holy Spirit alone, and there is no retreat to any secondary ground for the authority of the Bible whether it may be miracles, the victory of Church History, rational cohesiveness, etc.
[^Header Background Image Source] By signed "Ergotimos made me; Kleitias painted me” (Florence Archaeology Museum) [Public domain], via Wikimedia Commons
[^1] Barth, Karl. Church Dogmatics, Vol. 1.2, Sections 19-21: The Doctrine of The Word of God. Trans. T. F. Torrance. Vol. 5. London: T & T Clark, 2010. 535-37. Print. Study Edition. (paragraph breaks and indents added for readability, and Latin has been replaced translation in italic text from study edition footnotes).