Below I have included the notes that I prepared for this first meeting. It is basically a summary of what Barth does in these two paragraphs. Hopefully they will be interesting and / or helpful in to someone in some way. Also, if you are in the Princeton area and would like to attend the reading group, feel free to contact me.
§ 1. The Task of Dogmatics
As a theological discipline dogmatics is the scientific self-examination of the Christian Church with respect to the content of its distinctive talk about God.
1. The Church, Theology, Science
Theology is tied to the unity of the church’s being and act, even while the church’s act does not coincide with its being. The confession of the church is both a confession of God and the confession of the need for responsible human action. (3)
Theology can be spoken of in three ways: as a science, as testimony and as service. Barth is concerned with the former, which has to do with the church’s critical reflection on its talk about God. It is meaningful as obedience to grace. The criterion by which theology (as science) evaluates the church is the church’s being, namely, Jesus Christ. (4)
Biblical theology = basis, practical theology = goal, dogmatic theology = content. Theology is not a science to the exclusion of other sciences, which only create the need for theology as a distinct science by their inattention to the object of theology (God – cf. the editor’s preface where it is noted that God as theology’s ‘object’ is a concession to the translation task; in German there is more of a sense of an acting subject). (5)
The other sciences have tried to critique the church but it is necessary to do so on the basis of the church’s being (Jesus Christ). Since the other sciences do not use this criterion, theology as a distinct science is necessary. (6-7)
Calling theology a ‘science’ means  it is concerned with “a definite object of knowledge,”  it has a definite and consistent methodology,  it must give an account of this methodology. The existence of other sciences spurs theology on to greater faithfulness to itself, but it does not have to justify itself before the other sciences. (8-9)
In calling itself a science theology does not submit to the standards of the other sciences. Theology as a science is not systematically related to the other sciences because this would be admitting that theology as a distinct science is a necessary thing rather than an emergency measure. Theology proves itself as science by doing its own work. (10)
Theology should be called a science because  recognition of other sciences engenders humility,  it protests against the general concept of a science which is pagan,  theology shows that it does not take the heathenism of other sciences seriously enough to withdraw from them. (11)
2. Dogmatics as an Enquiry
 Dogmatics as enquiry presupposes that God can be known. This on the basis of Jesus Christ, the criterion of dogmatics which is given in completeness. (This is the analogia fidei.) The second event of human dogmatics or speech about God is united to but distinct from this first act (sacramental language employed). (12-13)
 Dogmatics as enquiry presupposes that the true content of speech about God must be known. It must conform to Christ, but this conformity is never unambiguous. Though the divine answer is perfectly given, dogmatics continues in the form of a question. “[T]he creaturely form which the revealing action of God assumes in dogmatics is never that of knowledge attained in a flash…but a laborious movement from one partial human insight to another…” This is the meaning of theologia crucis. Humility. The notion of church as divine and human reality supports all this. (14-15)
Dogmatics does not just repeat the prophets, apostles, creeds, etc, but asks what we must say today on the basis of these things. (16)
3. Dogmatics as an Act of Faith
Dogmatics is the work of human knowledge but human knowledge under the decisive condition of faith. What is faith? Faith is obedience to the call of Christ, and therefore faith knows God. It is only through such faith that human action (knowing?) is related to the being of the church (Jesus Christ). Faith, however, is not a determination that one can give to one’s own activity, but it is always a matter that rests with God. “[I]f we say that dogmatics presupposes faith…we say that at every step and with every statement it presupposes the free grace of God which may at any time be given or refused…” (Small print on Anslem, Schleiermacher, etc. – avoid anthropologizing theology.) The church must act and do theology, but whether it does so in accordance with its being is up to God. (17-21)
Because of all this, dogmatics must be done as humble, repentant obedience. The act of faith makes dogmatics possible and calls it into question. (Predestination is mentioned as a way of basing faith in God’s activity.) we are thus cast upon the necessity of prayer. “Lord, I believe; help thou mine unbelief.” (22-24)
§ 2. The Task of Prolegomena to Dogmatics
Prolegomena to dogmatics is our name for the introductory part of dogmatics in which our concern is to understand its particular way of knowledge.
1. The Necessity of Dogmatic Prolegomena
It is said that prolegomena to theology is particularly necessary presently, and that it should be conducted in terms of a ‘point of contact’ in the human person for the Gospel. This is to be rejected:  because there is no theological reason to assume that times are different now and thus require this procedure, revelation has always contended with unbelief,  to proceed in this way would be to abandon the serious task of dogmatics for it puts the resistance of human reason to the Gospel at the center,  though this procedure claims to make theology relevant it actually makes it less so because dogmatics is relevant to the extent that it carries out its unique task and [a] in this proposed procedure faith must take unbelief so seriously as to not take itself seriously enough, [b] dogmatic work is presumed to be complete, [c] if this battle is won then dogmatics will likely become complacent. “Apologetics and polemics can only be an event and not a programme.” (25-31)
Faith is not primarily in conflict with unbelief, but with itself, insofar as it recognizes that it possesses the potential for unbelief within itself and expresses itself as such. This happens especially with reference to heresy for “In this conversation the Church must wrestle with heresy in such a way that it may itself be the Church. And heresy must attack the Church because it is not sufficiently or truly the Church,” that is, what is taken for faith is confronted with the possibility that it is actually unbelief. (31-33)
Barth specifically has in view Roman Catholicism and “pietistic and rationalistic Modernism,” against which “the Evangelical Churches” must position themselves. In each case the church must stand against itself and against the possibilities that its own faith possesses. This has been a special problem since the time of the Reformation, and the Scripture principle has been important in sorting out where the Evangelical churches stand with reference to both of these antagonists. (34-36)
2. The Possibility of Dogmatic Prolegomena
Prolegomena since the enlightenment has been done by establishing a general ontology or anthropology to ground the claims that will be made. Schleiermacher and Schweizer come up in fine print. “The assertion of an existentially ontological prius to ontically existential faith, or the definition of faith as a mode of the historical being of human existence, is a cardinal proposition of the faith which understands the being of the Church and itself decisively as a definition of the reality of man, of piety.” There is a fairly lengthy (at least judged by the text thus far) fine print section on Heirich Barth and his overcoming of the philosophical trends that this sort of prolegomena relied upon. (36-40)
On the other hand, the Catholics do things differently and end up trading Jesus’ lordship for the Church’s existence. Analogia entis. Countering, the being of the church (remember, Jesus Christ) must be understood as actus purus, which means that he is other than the church and that dogmatics can be done only in the moment of his speaking and our hearing. (40-41)
Thus, prolegomena must be done as part of dogmatics. Dogmatic knowledge can only be an event and thus is not grounded in “general human possibility or an ecclesiastical reality.” Prolegomena is, therefore, when theologians “give an account of the way which we tread.” Basically and / or practically, prolegomena means establishing the Scripture principle, but not Scripture in a restrictive sense but as subordinate to a more robust account of the Word of God.
 In GD Barth speaks of sciences in general, and of theology as science, thus: “I myself would equate ‘scientific’ and ‘objective,’ objectivity being the closest possible adjustment of knowledge to the distinctiveness of its object. The most fitting means to establish objective truth, the most certain way to achieve coherence in knowledge, the freest critical norm, and the most logical grounding of all knowledge will in ever field of science be the truth itself, which we do not have to produce but which is given to us. If, then, the means of knowledge, the coherence, the criticism, and the grounding must be determined by the distinctiveness of the object in dogmatics, this does not preclude its scientific character but includes it.” (8)
 A similar notion appears in GD when Barth writes: “But there comes a point – and no theologian can evade it – at which theology does become dangerous and suspect. This is the point where the twofold question arises: What are you going to say? Not as one who knows the Bible or Thomas or the Reformers or the older Blumhardt, but responsibly and seriously as one who stands by the words are are said: you? And what are you going to say?…You? What? These are the questions of dogmatics. We have to consider the fact that ‘in some way’ we have to speak about God.” (6)
 Cf. Barth’s discussion of the relation between church and Christ, as well as the relation between the communio sanctorum and the communio peccatorum, in CD IV/2 (614-726).