I think it is valuable to look at the relation of Badiou and Kant through the lens of Lacan's concept of "logical time". A number of writers have pointed out affinities between Kant and Badiou but Badiou himself has situated his thought as actively antagonistic to Kant. One possible correspondance which I have not yet seen discussed is the relation of Kant's domains (scientific, ethical, aesthetic) to Badiou's generic procedures (art, science, politics, love). I think that by comparing them the kernel of difference between the two philosophers could be made more visible and it could also help clear up an ambiguity in the work of both.
As I understand it (and this understanding is based almost entirely on secondary sources, specifically Hallward), Badiou constructs the four generic procedures as the four ways individual can relate to collective in the articulation of a truth. So there is truth articulated in the radical isolation of a loving couple, the collective subject which emerges from the declaration of a new humanity, the subjective experience of art directed at a potentially uninversal public, and the singular experience of the epistemological break of the scientific discovery also direced at the universal public. There is an ambiguity here: under the sole dimension of individual/collective one cannot make a distinction between artistic truths which appear as the name of the void rupturing the materiality of language and the scientific truth of the epistemological break. I admit I am still a novice to Badiou's thought, so any clarification of this point is welcomed.
In Kant, as I touched on in my last post, there is a similar figure of disjunct spaces of truths. The domains of aesthetics, ethics, and science are not quite as clearly defined in Kant as Badiou's domains are, and Kant also does not presume to exhaust the modalities of truth through his domains (my knowledge here is again secondary, so bear with me). Still, it is clear that Kant would hold each of these domains as separate only through their mutual bracketing and thus that their respective fields of study would only come to exist as radically exclusive of each other. Kant differentiates his domains based on three criteria by which a spectator can validate an instance of judgement pronounced by a subject. Pleasure/displeasure in the domain of aesthetics, good/bad in the domain of morality, and true/false in the domain of science.
My question for both Badiou and Kant is how does one pass from one domain to the next? It is here that the idea of logical time deserves to be brought up as it offers a way of thinking about not only the event/actions which occur within each domain but a potential method of traversing domains which could provide an answer to the question of whether they are bounded/unbounded (though we know from Kant that unbounded does not mean infinite).
To briefly recount the logical anecdote described by Lacan in "Logical Time and the Assertion of Anticipated Certainty": there are three prisoners who are told that they each have a dot on their back colored either white or black and that there are five total dots, three white and two black. The first prisoner to correctly deduce what color is on their own back with a logical justification is granted freedom.
Here's how it goes down: the first prisoner sees two whites and knows that there are only three. He is unable to come to a conclusion immediately and so he is forced to imagine himself as one color or the other and extrapolate as to the effects that would produce on the decision making process of the other prisoners. He speculates that he is black. If he were black, then the second prisoner would see one black and one white and would also be forced speculate. Prisoner two could then say to himself, "If I were black, then prisoner three would leave immediately, prisoner three does not leave immediately so therefore I am white" and he would then be able to leave. He does not. Finally the circuit of logic is returned to the chosen First prisoner. What makes this diaphonous little puzzle important for both Lacan and Badiou is that the action which happens at this point is not simply a matter of course. Prisoner One must realize that his reasoning only makes sense if the fact that the other prisoners haven't moved is a fact of their hesitation, and not merely slow-thinking. In other words, Prisoner One must become Prisoner One in his act of stepping forward, the logic can only be applied retroactively.
The reason I recount this is to propose the following: What if this logical time, with its three necessary steps, was taken as a temporal architechtonic in place of Hegel's master-slave dialectic? With only two players, the dialectic necessarily produces a subject to a law forever split in two (to search forever in vain for the dead master of pure recognition) and whose revolution must always miss the Real. Hegel's abundant critics have pointed out how the mechanism of dialectic ends up priveleging a resignation to natural law, even if the law is shown to be an always riven and always elusive entity.
But is the only other alternative to imagine the time of Truth as always singular and instantaneous (as Badiou does)? Of there being no iterable logic of time which could formally connect different modalities of truth (i.e. artistic, mathematical, ethical, familial, pedagogical)? Logical time, in its preservation of the irreducibly unjustifiable Real moment of the act could provide such a principle. This could provide somewhat demystify the process by which a Truth comes into being, the path it must travel from one register to the next.
This all highly speculative, I really don't know if I agree with what I've said, but at least it is now said.