Logics of Worlds

Being and Event II

Alain Badiou, Alberto Toscano

• Publisher: Bloomsbury Academic
• Year: 2019
• ISBN:
  9781350043022
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Logics of Worlds is the sequel to Alain Badiou's masterpiece, Being and Event. Tackling the questions that had been left open by Being and Event, and answering many of his critics in the process, Badiou supplements his pioneering treatment of multiple being with a daring and complex theory of the worlds in which truths and subjects make their mark - what he calls a materialist dialectic. Drawing on his most ambitious philosophical predecessors - Leibniz, Kant, Hegel, Kierkegaard, Lacan, Deleuze – Badiou ends this important later work with an impassioned call to 'live for an Idea'.

BOOK II

GREATER LOGIC, 1. THE TRANSCENDENTAL

INTRODUCTION

Book II is entirely devoted to a single concept, that of the transcendental. The word ‘transcendental’ is warranted here because it encapsulates my recasting, with respect to Being and Event, of the primitive notion of ‘situation’, replaced here by that of ‘world’. Where the earlier book followed the thread of ontology, my current undertaking, placed under the rubric of the transcendental, unravels the thread of logic. Previously, I identified situations (worlds) with their strict multiple-neutrality. I now also envisage them as the site of the being-there of beings. In Being and Event, I assumed the dissemination of the indifferent multiple as the ground of all that there is, and consequently affirmed the ontological non-being of relation. Without going back on this judgment, I now show that being-there as appearing-in-a-world has a relational consistency.

I have established that ‘mathematics’ and ‘being’ are one and the same thing once we submit ourselves, as every philosophy must, to the axiom of Parmenides: it is the same to think and to be. It is now a matter of showing that ‘logic’ and ‘appearing’ are also one and the same thing. ‘Transcendental’ names the crucial operators of this second identity. Later, we will see that this speculative equation greatly transforms the third constitutive identity of my philosophy, the one which, under the sign of evental chance, makes ‘subject’ into a simple local determination of ‘truth’. It effectively obliges us to mediate this determination through an entirely original theory of the subject-body.

The first three sections of Book II follow the rule of triple exposition: conceptual (and exemplifying), historical (an author) and formal. The substance of what is presented three times can also be articulated in terms of three motifs: the necessity of a transcendental organization of worlds; the exposition of the transcendental; the question of what negation is within appearing.

Since I declare that ‘logic’ signifies purely and simply the cohesion of appearing, I cannot avoid confronting this assertion with the astounding fortunes of logic in its usual sense (the formal regulation of statements) among those philosophies which believed they could turn the examination of language into the centre of all thought, thereby consigning philosophy to fastidious grammatical exercises. It’s no mystery that in the final analysis this is a matter of bringing philosophy into the space of university discourse, a space which conservatives of every epoch have always argued it should never have left. Today, it is also quite clear that if we allow ourselves to be intimidated, philosophy will be nothing more than a scholastic quarrel between liberal grammarians and pious phenomenologists. But when it comes to logic I do not content myself with political polemic. After all, I admire the great logicians, those whom—like Gödel, Tarski or Cohen—carried out a commendable mathematical incorporation of the inherited forms of deductive fidelity. I will also carefully establish that logic, in its usual linguistic sense, is entirely reducible to transcendental operations. This will be the object of Section 4 of this Book.

We will then see—it is the object of Section 5—that the simple consideration of transcendental structures allows us to define what a classical world is, that is a world which obeys what Aristotle already considered as a major logical principle: the excluded middle. This principle declares that, given a closed statement A, when we interpret this statement in a world we necessarily have either the truth of A or that of non-A, without any third possibility. A world for which this is the case is a classical world, which is an entirely particular ‘case of world’. Without going into the details, I will show that the world of ontology, that is the mathematics of the pure multiple, is classical. In the history of thought up to the present day, this point has had truly innumerable consequences.

The ‘historical’ companion of this Book is Hegel, the thinker par excellence of the dialectical correlation between being and being-there, between essence and existence. We will be measuring ourselves up to his Science of Logic.

Let’s now quickly present what is at stake in each of three themes that make up the main content of the first three sections.

1 Necessity of a transcendental organization
of the situations of being

The first section consists of a long demonstration. It is a matter of forcing thought to accept that every situation of being—every ‘world’—far from being reduced to the pure multiple (which is nonetheless its being as such) contains a transcendental organization.

The meaning of this expression will transpire from the demonstration itself. As in Kant, we are trying to resolve a problem of possibility. Not however ‘how is science possible’ or ‘how are synthetic judgments a priori possible’ but: how is it possible that the neutrality, inconsistency and indifferent dissemination of being-qua-being comes to consist as being-there? Or: how can the essential unbinding of multiple-being give itself as a local binding and, in the end, as the stability of worlds? Why and how are there worlds rather than chaos?

As we know, for Kant the transcendental is a subjectivated construction. With good reason, we speak of a transcendental subject, which in some sense invests the cognitive power of empirical subjects. Ever since Descartes, this is the essential trait of an idealist philosophy: that it calls upon the subject not as a problem but as the solution to the aporias of the One (the world is nothing but formless multiplicity, but there exists a unified Dasein of this world). The materialist thrust of my own thought (but also paradoxically of Hegel’s, as Lenin remarked in his Notebooks) derives from the fact that within it the subject is a late and problematic construction, and in no way the place of the solution to a problem of possibility or unity (possibility of intuitive certainty for Descartes, of synthetic judgments a priori for Kant).

The transcendental that is at stake in this book is altogether anterior to every subjective constitution, for it is an immanent given of any situation whatever. As we shall see, it is what imposes upon every situated multiplicity the constraint of a logic, which is also the law of its appearing, or the rule in accordance with which the ‘there’ of being-there allows the multiple to come forth as essentially bound. That every world possesses a singular transcendental organization means that, since the thinking of being cannot on its own account for the world’s manifestation, the intelligibility of this manifestation must be made possible by immanent operations. ‘Transcendental’ is the name for these operations. The final maxim can be stated as follows: with regard to the inconsistency of being, ‘logic’ and ‘appearing’ are one and the same thing.

However, it does not follow, as in Kant, that being-in-itself is unknowable. On the contrary, it is absolutely knowable, or even known (historically-existing mathematics). But this knowledge of being (onto-logy) does not entail that of appearing (onto-logy). It is this disjunction which the arguments in this section attempt to force. The stages which develop this parameter have as their point of departure the impossibility of determining a being of the Whole, and finally the thesis according to which there is no Whole. Contrary to a Heideggerian proposition, it is irrational to evoke ‘beings-as-a-whole’. It follows that every singular being [étant] is only manifested in its being [être] locally: the appearing of the being of beings [l’être de l’étant] is being-there. It is this necessity of the ‘there’ which, for a being thought in its multiple-being, entails a transcendental constitution (without subject). This constitution authorizes us to think the being as localized, to include the ‘there’ in the thinking of being—something that the mathematical (ontological) theory of the pure multiple, despite conveying the whole being of the being, does not allow. In what follows, we will call universe the (empty) concept of a being of the Whole. We will call world a ‘complete’ situation of being (this will be gradually elucidated). Obviously, since we show that there is no universe, it belongs to the essence of the world that there are several worlds, since if there were only one it would be the universe.

2 Exposition of the transcendental

‘Exposition of the transcendental’ signifies the description of the logical operators capable of lending coherence to appearing ‘in’ one of the worlds in which multiples come to be.

I write ‘in’ a world (in quotation marks) to indicate that we are dealing with a metaphor for the localization of multiples. As a situation of being, a world is not an empty place—akin to Newton’s space—which multiple beings would come to inhabit. For a world is nothing but a logic of being-there, and it is identified with the singularity of this logic. A world articulates the cohesion of multiples around a structured operator (the transcendental).

At the core of transcendental questions lies the evaluation of the degrees of identity or difference between a multiple and itself, or between a being-there and other beings. The transcendental must therefore make possible the ‘more’ and the ‘less’. There must exist values of identity which indicate, for a given world, to what extent a multiple-being is identical to itself or to some other being of the same world. This clearly requires that the transcendental possess an order-structure.

The other operations necessary to the cohesion of multiples in a world constitute a minimal phenomenology of abstract appearing, by which we mean what is conceptually required for appearing to be bound. We are speaking here of any appearing whatsoever in any world whatsoever. In other words, our operational phenomenology identifies the condition of possibility for the worldliness of a world, or the logic of the localization for the being-there of any being whatever.

It is very striking that this phenomenology is absolutely complete with only three operations, bound together by a simple axiomatic:

a.A minimum of appearance is given.

b.The possibility
of conjoining the values of appearance of two multiples (and therefore of any finite number of multiples).

c.The possibility of globally synthesizing the values of appearance of any number of multiples, even if there is an infinity of them.

When we say that the description of these operations makes up a complete phenomenology, we mean that the transcendental determination of the minimum, the conjunction and the envelope (or synthesis) provides everything that is needed for being-there to consist as a world.

Upon reflection, this economy of means is natural. Appearing is undoubtedly bound (and not chaotic) if I can, first, set out degrees of appearance on the basis of non-appearance (minimum); second, know what two multiples do or do not have in common when they co-appear (conjunction); and finally, if I am capable of globalising appearance in such a way that it makes sense to speak, in a given world, of a region of appearing (envelope). The form of appearing is accordingly homogeneous, so that the reciprocity between logic and appearing, and between consistency and being-there, is well-founded.

The exposition of the transcendental will thus proceed through three successive moments. The first forces upon thought the existence, in the transcendental, of a minimum. The second (finite operational domain) fixes the laws relative to the conjunction of two beings-there. Finally, the third (infinite operational domain) posits the existence of an envelope for every region of the world. A supplementary step will concern not the operations that lend consistency to appearing, but rather the consistent appearing of these operations themselves.

Keeping in mind the exceptional importance of the notion of causal bond, or necessity, in the transcendental tradition, we will conclude by showing that it is possible to derive from the three fundamental operations a measure of the correlation in a given world between the appearing of a multiple and that of another multiple. We will call this correlation dependence. In terms of the logical status of appearing, dependence interprets in a world the logical relation of implication.

3 The origin of negation

In every transcendental theory of being-there, the question of negation is very complex. This is testified to in particular by the very dense passage that Kant devotes to the conceptualization of the ‘nothing’ in the Critique of Pure Reason, at the very end of the ‘Transcendental Analytic’. Kant claims that as such the question ‘is not in itself especially indispensable’. In effect, he makes do with a page and a half, opening without delay onto the Dialectic, after an absolutely impenetrable formula: ‘Negation as well as the mere form of intuition are, without something real, not objects’. This suggests that appearing or phenomenality—in which everything presents itself to intuition under the form of the object—knows no negation. But the question is really far more complex. Kant has little difficulty in identifying negation on the side of the concept: there is the ‘empty concept without object’, which is to say that which, albeit coherent for thought, does not relate to any intuitable object, for example, the Kantian notion of the noumenon. And there is the concept that contradicts itself, like the square circle, which makes any object impossible. In these two cases, the concept points to the absence of any object: nothing appears, there is nothing there. Or, as in the case of the word ‘noumenon’, nothing is there but being without being-there, being out of the world. Except that Kant also wishes to extend the question of the negative to intuition, which is where things get rather awkward. He isolates the ‘pure form of intuition’ in its empty exercise, or designated as empty, which is to say ‘without object’. This goes for the a priori forms of sensibility: time and space. Must we accord to these forms a trans-phenomenal real? Does Kant want to say that the forms of objectivity ar...