Continuing from part one on Conditions (1992), Ch. 1 "The (Re)turn of Philosophy Itself." What makes philosophy possible? Well, there are truths, and these come out of the four "conditions," i.e. mathematics (and science more generally), politics, art, and love. This precludes skepticism about truth. Because there are many truths, Badiou's philosophy is pluralistic, but these truths can be combined in thought, i.e. into a philosophy, so in that sense Truth is One, not many. This doesn't mean that you can gather all the truths together into one Truth and then be a dogmatic jerk about it. What philosophy does instead in this "bringing together" is not altogether clear, given that he doesn't want to be strictly negative and skeptical (this seems to be Wes' reading). Perhaps the point is pragmatic and humanist: Philosophy somehow makes these truths live for us, and so makes us more lively and human.
A possible connection to make here is to Heidegger's "humanism," which identified the purpose of philosophy as "disclosing the clearing of Being," in other words removing us from the mass of the everyday ("knowledge" for Badiou) so that we can confront and react to the truly novel and authentic, i.e. truths as emergent from what Badiou calls events. Philosophy provides a reflective space (an "interval" or "void") that provides perspective on the individual truths different from what the person actually falling in love or engaged in a scientific discovery or creating a work of art would have.
We get into more of Badiou's "pincer" metaphor for philosophy: One of the arms is the "regulating of successiveness by argument" (i.e. reason), and the other is "the declaration at a limit." So philosophy on the one hand breaks out of the regularity of mere knowledge, i.e. inverts the succession of events that make up ordinary life. In this sense it is a "fiction of knowledge" as Badiou puts it. Saying that (for instance, as Descartes does) that mind and body are fundamentally separate ontological entities is a different kind of claim than saying something empirical about the relations that we note between mind and body in either an everyday setting (like that I will my arm to move and it does) or a scientific setting (like when a subject reports thinking of his mother and the neurologist notes a correlated activation of certain neurons). The ontological claim is not scientific; it can't be verified or refuted by experiment. It's like knowledge, but not.
The other side of the pincer is the limit, and maybe this is like the Kantian limit between the phenomena that science studies and the noumena, which are the way things are in themselves apart from normal knowledge. The ontological claim (or any philosophical claim) is going beyond the limits of normal discourse and knowledge. This is a "fiction of art," according to Badiou's language, in that I'd be using metaphor instead of argument to describe something transcendent, just like art does.
So the two pincers are rational argumentation and artistic imagination, seizing the truth and so making it seize us, i.e. we're dazzled by it, like we're having a seizure. (Badiou refers to that double meaning of "seizure" there: grabbing and having a fit. Presumably the double meaning is retained in French!) So Truth is not a foundation of truths, but the "pincers of Truth" enable us to grasp and behold truths.
Our official reading also included chapters 2 and 3 of the book. While ch. 2 is really just a short summary of his theory of truth (a recap), ch. 3, "What Is a Philosophical Institution," is about the what kind of academic organization a philosophy department should be given his idiosyncratic definition of philosophy. It should not be dogmatic, dedicated to reigning in the excesses of its faculty. It should serve as a "knot" to provide a unity of purpose: It should be open to all students, all about connecting thinkers with disciples, and a publishing house. The key here is the difference between multiplicity (which Badiou agrees with) and relativity (which he does not). The community of thinkers needs to be engaging with each other, not a hands-off "anything goes" plurality of "diverse" thinkers.