Badiou starts with four fundamental theses on truth:
1) Starting from Hiedeggar, there is no other solution to the question of truth than that of the poem. While, as John noted, Badiou might seem similar to Hiedeggar, he is actually opposed to him. B. argues that the for H., only the poem can properly reveal Being, and also our being-in-the-world (this movement to the poem, while latent in B&T is in full blown in the later writings). For H., Dasein is always already in the 'truth' of being. Before the 'truth' of assertions, Dasein is in the Truth, the primordial Truth of Being of which he falls into forgetfulness through predication/metaphysics. But equally, Dasein is in the un-truth, or is in erring. But for Badiou, we are never simply in the 'truth' because truths are new and unexpected. For Badiou, Truth is not a structure of our being, but something that becomes. And the best way to speak of this newness of Truth is not through the poem (as for Heideggar) but through mathematics (a mathematics of which H. would have no part).
2) But to move beyond the poem we must also distinguish truth from the narrow form of proposition and judgment of the analytic tradition. So truth is neither the poem nor proposition.
3) so “we must conceive of a truth both as the construction of a fidelity to an event, and as the generic potency of a transformation of a domain of knowledge” (43).
4) In light of the above, the essential categories of truth are negative: undecidablity, indiscernibiltiy, the generic not-all, and the unnameable.
After passing through a couple comments on Hiedeggar’s concept of truth, Badiou begins his own ideas, begin with the distinction between knowledge and truth, with truth as a becoming of something new. It is worth quoting a lengthy section:
“I will start from the following idea: a truth is, first of all, something new. What transmits, what repeats, we shall call knowledge. Distinguishing truth from knowledge is essential. It is a distinction that is already made in the work of Kant: the distinction between reason and understanding. It is a capital distinction for Heidegger: the distinction between truth—alethia—and cognition or science—techne.” (45)If a truth is distinguished from knowledge, as the becoming of something new, then what is the process of its appearing? “A truth must be sumitted to thought, not as a judgment, but as a process in the real” (45). This begins an examination of what Badiou calls the “truth process,” filling out the four negative of truth.
The process of a truth begins when something happens, where there is an event. Knowledge is a repetition of what we already know, of common sense and received knowledge. Truth is a supplement to knowledge, and knowledge is always of what exists, of what is, so Truth is something that happen as a becoming beyond what already is. This is initiated in the event. The event is the first negative of truth: the undecidable, because the realm of knowledge cannot decide if this thing called the event actually exists. But the undecidability of the event is such that it induces a subject, a subject which decides that the event did indeed happen, that it does exist, and this subject is constituted in this act of decidinig to be faithful to the event. Fidelity is the nature of the subject of the event, through the decision, is caught between two indiscernibles. Two concepts are indiscernible when no language game can distinguish between them. But the subject of truth, does distinguish between them based on his fidelity to the event. From the faithful discernment of the Subject, between indiscernible concepts, the process of the truth continues, producing a generic subset within the previous situation of knowledge. The generic subset can never be fully named, otherwise it would fall under the rule of knowledge and under the sway of mere existence. The process of truth in its becoming can never be fully realizing, and therefore always has the unnameable as its limit. In fact, for Badiou, the forcing of a truth into total nomination, is the basis of Evil.