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Alain Badiou “Logics of Worlds” (III & IV raamat)

Badiou, Alain 2009 [2006]. Logics of Worlds. Being and Event, 2. London, New York: Continuum.

Book III. Greater Logic, 2. The Object.

This book proposes an entirely new concept of what an object is. We are obviously dealing with the moment of the One in our analysis. That is because by  ‘object’ we must understand that which counts as one within appearing, or that which authorizes us to speak of this being-there as inflexibly being ‘itself’. The main novelty of this approach is that the notion of object is entirely independent from that of subject. (193)

So our trajectory can be summed up as follows: (object-less) subjective formalism, (subject-less) object and objectivity of the subject (bodies). It inscribes into the logic of appearing the generic becoming of truths which Being and Event had treated within the bounds of the ontology of the pure multiple. (194)

Since we posit that appearing has nothing to do with a subject (whether empirical or transcendental), naming instead the logic of being-there, we clearly cannot oppose an inner to an outer experience. In fact, no experience whatsoever is involved. But we are obliged to establish that an object is indeed the being-there of an ontologically determinate being; or that the logic of appearing does not exhaustively constitute the intel-ligibility of objects, which also presupposes an ontological halting point that is at the basis of appearing as the determination of objects-in-the-world. (195)

[…] being itself may, under certain conditions, be synthesized (enveloped), and therefore be ascribed a unity other than the one that counts its pure multi-plicity as one. Everything happens as if appearance in a world endowed pure multiplicity—for the ‘time’ it takes to exist in a world—with a form of homogeneity that can be inscribed in its being. This (demonstrable) result—which shows that appearing infects being to the extent being comes to take place in a world—is so striking that I have named it the ‘fundamental theorem of atomic logic’. (196-197)

We have called the ‘phenomenon’ of a multiple-being, relative to the world in which it appears, the giving of the degrees of identity that measure its relationship of appearance to all the other beings of the same world (or, more precisely, of the same object-of-the-world). This definition is relative and by no means rests, at least in an immediate sense, on the intensity of appearance of a being in a world. (207)

Given a world and a function of appearing whose values lie in the transcendental of this world, we will call ‘existence’ of a being x which appears in this world the transcendental degree assigned to the self-identity of x. Thus defined, existence is not a category of being (of mathematics), it is a category of appearing (of logic). In particular, ‘to exist’ has no meaning in itself. In agreement with one of Sartre’s insights, who borrows it from Heidegger, but also from Kierkegaard or even Pascal, ‘to exist’ can only be said relatively to a world. In effect, existence is nothing but a transcendental degree. It indicates the intensity of appearance of a multiple-being in a determinate world, and this intensity is by no means prescribed by the pure multiple composition of the being in question. (208)

We will now establish a fundamental property of existence: in a given world, a being cannot appear to be more identical to another being than it is to itself. Existence governs difference. (210)

That existence subsumes difference (through its transcendental degree) does not make existence into the One of appearing. The fact that existence is not a form of being does not make it into the unitary form of appearing. As purely phenomenal, existence precedes the object and does not constitute it. (211)

Given a world, we call object of the world the couple formed by a multiple and a transcendental indexing of this multiple, under the condition that all the atoms of appearing whose referent is the multiple in question are its real atoms.

In an abstract sense, it should be underscored that an object is jointly given by a conceptual couple (a multiple and a transcendental indexing) and a materialist prescription about the One (every atom is real). It is therefore neither a substantial given (since the appearing of a multiple A presupposes a transcendental indexing which varies according to the worlds and may also vary within the same world) nor a purely  fictional given (since every one-effect in appearing is prescribed by a real element of what appears). (220)

In a general sense, we will call  ‘localization of an atom on a transcendental degree’ the function which associates, to every being of the world, the conjunction of the degree of belonging of this being to the atom, on one hand, and of the assigned degree, on the other.

It appears then that every assignation of an atom to a degree—every localization—is itself an atom. Mastering the intuition of this point is both very important and rather difficult. In essence, it means that an atom which is ‘relativized’ to a particular localization gives us a new atom. (224)

Let us restate this definition more explicitly: Take an object presented in a world. Let an element ‘a’ of the multiple ‘A’ be the underlying being of this object. And let ‘p’ be a transcendental degree. We will say that an element ‘b’ of A is the ‘localization of a on p’ if b prescribes the real atom resulting from the localization on the degree p of the atom prescribed by element a. (225)

At this point in our discussion, it is very important to emphasize once again that localization is a relation between elements of A, and therefore a relation that directly structures the being of the multiple. (225)

At the point of a real atom, being and appearing conjoin under the sign of the One. It only remains to formulate our  ‘postulate of materialism’, which authorizes a definition of the object. As we know, this postulate says: every atom is real. It is directly opposed to the Bergsonist or Deleuzean pre-supposition of the primacy of the virtual. In effect, it stipulates that the virtuality of an apparent’s appearing in such and such a world is always rooted in its actual ontological composition. (250-251)

It is crucial to remember that existence is not as such a category of being, but a category of appearing; a being only exists according to its being-there. And this existence is that of a  degree of existence, situated between inexistence and absolute existence. Existence is both a logical and an intensive concept. It is this double status which makes it possible to rethink death. (269)

Just like existence, death is not a category of being. It is a category of appearing, or, more precisely, of the becoming of appearing. To put it otherwise, death is a logical rather than an ontological concept. All that can be affirmed about ‘dying’ is that it is an affection of appearing, which leads from a situated existence that can be positively evaluated (even if it is not maximal) to a minimal existence, an existence that is nil  relatively to the world. (269-270)

[…] what comes to pass with death is an exterior change in the function of appearing of a given multiple. This change is always imposed upon the dying being, and this imposition is contingent. The right formula is Spinoza’s:  ‘No thing can be destroyed except through an external cause’. It is impossible to say of a being that it is ‘mortal’, if by this we understand that it is internally necessary for it to die. At most we can accept that death is possible for it, in the sense that an abrupt change in the function of appearing may befall it and that this change may amount to a minimization of its identity, and thus of its degree of existence. (270)


Book IV. Greater Logic, 3. Relation

[…] a relation is a connection between objective multiplicities—a function—that creates nothing in the register of intensities of existence, or in that of atomic localizations, which is not already prescribed by the regime of appearance of these multiplicities (by the objects whose ontological support they are). It is on this basis that the question of the universality of a relation poses itself. We will say that a relation is universally exposed in a world if it is clearly ‘visible’ from the interior of this world, in a sense which will be specified below. (301)

[…] a relation is universal if its intra-worldly visibility is itself visible. It is then effectively impossible to cast doubt on its existence. Within the full extension of the world, it is a relation for all. These considerations allow us to establish one of the most striking results of the analytic of worlds. We will demonstrate that every relation is uni-versal. More precisely, we will demonstrate that the infinity of a world (its ontological characteristic) entails the universality of relations (its logical characteristic). The extensive law of multiple being subsumes the logical form of relations. Being has the last word. It already did at the level of atomic logic, where we affirmed, under the name of ‘postulate of material-ism’, that every atom is real. That is why the universality of relations—which is itself not a postulate but a consequence—is accorded the status of ‘second constitutive thesis of materialism’. (302)

[…] we must think two types of relations, in order to secure an intelligible answer to the question of what a world is:

a. the constitutive relations (or operations) of the theory of the pure multiple, or theory of being-qua-being; in effect, every world is con-structed on the basis of multiple-beings, and it is important to know under what conditions these multiple-beings are globally exposed to constituting the being of a world;

b. the relations between apparents of the same world, that is to say the relations between objects. (305)

[…] if you totalize the parts of an (ontological) component of a world, counting as one the system of these parts, you get an entity of the same world. This is the second fundamental property with regard to the operative extension of a world thought in its being: a world makes immanent every local totalization of the parts of that which com-poses it. Its state (the count as one of the subsets of the beings that are there) is itself in the world, and not transcendent to it. Just as there is no ultimate formless matter, so there is no principle of the state of affairs. (308)

The extension of a world remains inaccessible to the operations that open up its multiple-being and allow it to radiate. Like the Hegelian absolute, a world is the unfolding of its own infinity. But, unlike that Absolute, the world cannot internally construct the measure or the concept of the infinite that it is. This impossibility is what assures that a world is closed, without it thereby being representable as a Whole from the interior of the scene of appearance that it constitutes. (309)

This paradoxical property of the ontology of worlds—their oper-ational closure and immanent opening—is the proper concept of their infinity. We will sum it up by saying: every world is affected by an inaccess-ible closure. (310)

A relation, within appearing, is necessarily subordinated to the transcen-dental intensity of the apparents that it binds together. Being-there—and not relation—makes the being of appearing. This is what we could call the axiom of relations. I say ‘axiom’ because of the intuitive, or phenomeno-logical, manifestness of its content: relation draws its being from what it binds together. The most rigorous formulation of the axiom could then be the following: a relation creates neither existence nor difference. Let’s recall that an object, the unit of counting of appearing, is the couple formed by a multiple-being and its transcendental indexing (or function of appearing) in a determinate world. We will then call ‘relation’ between two objects of a given world every function of the elements of the one towards the elements of the other, such that it preserves existences and safeguards or augments identities (that is, maintains or diminishes differences). (310)

A relation is an oriented connection from one object towards another, on condition that the existential value of an element of the first object is never inferior to the value which, through this connection, corresponds to it in the second object, and that to the transcendental measure of an identity in the one there corresponds in the other a transcendental measure which also cannot be inferior.

If we wish to move to a positive definition of relations, we will say: a relation between two objects is a function that conserves the atomic logic of these objects, and in particular the real synthesis which affects their being on the basis of their appearing. It is this definition in terms of conservation or invariants which we will adopt in the formal exposition. (312)

Every object—considered in its being as a pure multiple—is inexorably marked by the fact that in appearing in this world it could have also not appeared and, moreover, it may appear in another world. (321)

Generally speaking, given a world, we will call ‘proper inexistent of an object’ an element of the underlying multiple whose value of existence is minimal. Or again, an element of an apparent which, relative to the transcendental indexing of this apparent, inexists in the world. The thesis on the rationality of the contingency of worlds can then be stated as follows: every object possesses, among its elements, an inexistent. (322-323)

The inexistent of an object is suspended between (ontological) being and a certain form of (logical) non-being. We can conclude the following: given an object in a world, there exists a single element of this object which inexists in that world. It is this element that we call the proper inexistent of the object. It testifies, in the sphere of appearance, for the contingency of being-there. In this sense, its (onto-logical) being has (logical) non-being as its being-there. (324)

We will posit in effect that a functional connection between objects is identifiable as a relation only to the extent that it ‘conserves’ the principal transcendental particularities of these objects, in particular the degrees of existence and the localizations. This means that no relation has the power to upset the real atomic substructure of appearing. There is a resistance of matter. (336)

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