35 Propositions from Logiques des mondes



Alain Badiou

translated by Jake Bellone


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35 Propositions from Logiques des mondes


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There has been a book in the process of being written for the past three years. Its title is Logique des mondes. It goes its course, without haste, when I can. The first three chapters are nearly finished: The One, is “The Transcendental,” the Two “The Object,” the Three “The World.” After which things are going to change, to accelerate. We will leave the peace of analyses, the calmness of being. Damaged torments of subjectivation, true things, and no-things tout court. The Lights, in sum. Politics. Love. Science. Art. And thus in succession.

The first three chapters have a invariable mandate. An introduction, a conceptual exposition, a connection (a great classical author), a formal exposition, and finally the series of crucial propositions of the chapter. I offer here the propositions of these first three chapters. There are thirty-five of them. I leave them to their concentration; if not, I adjoin a glossary, intended in any event. These inevitable dictionaries are the gargoyles of this kind of cathedral.

Proposition 1: The All has no Being. Or: the concept of the universe is inconsistent.

Proposition 2: No being, except the Void, can be thought in its Being without recourse to Being already thought by at least one other being. In addition it is not the uniform procedure of this dependence.

Proposition 3: A being can only be thought as much as it inscribes itself into a world.

Proposition 4: To think a being such that it is inscribed into a world, or to think the Being-there of a being supposes the imposition of a logic of appearance which is not identical to the ontology (mathematical) of the pure Multiple.

Proposition 5: A logic of appearance, and therefore the logic of a world, returns to a unified failure of measure (intrinsic, without subject) of the identities and differences to the operations which depend on this measure. It necessarily pertains to a structure of order, giving sense to expressions like “more or less identical”, and, more generally, to comparisons of intensities. We name this order, and the operations which attach themselves, the transcendental of a situation (or of a world). The transcendental is designated by T, and the order which structures T by the customary symbol ≤. For a multiple-being, “appearance” means: to be seized by the logic of a world, therefore indexed to the transcendental of this world.

Proposition 6: The transcendental organization of any world allows the non-appearance of a being to be thought. This here means that it exists, in the structure of transcendental order, a minimal degree. We denote this μ.

Proposition 7: The transcendental organization of any world allows the evaluation of what is common in the Being-there of two beings which co-appear in this world. This supposes that, in the transcendental, given two degrees of intensity, there exists a third one which is “closest”, simultaneously, to the two others. This degree measures that which we name the conjunction or intersection of two beings-there. We denote this ∩.

Proposition 8: The transcendental organization of any world assures the cohesion of the Being-there of any part of this world. This supposes that to the degrees of appearance in this world of beings which constitute this part corresponds a degree which dominates all of them and at the same time is the smallest to do it. This degree, which synthesizes to the nearest point the appearance of a region of the world, is called the envelope of this region. If B is the region, the envelope of B is denoted ∑ B.

Proposition 9: In the order of appearance, the synthesis, global and in capacity of the infinite (the envelope), overrides the analysis, local and finite (the conjunction). Accordingly: the intersection of a singular appearance and of an envelope is itself an envelope. This is denoted: ∩ is distributive relative to ∑.

Proposition 10: In the order of appearance, there exists a transcendental measure of the degree of attachment necessary between two beings. We call this measure the dependence of one of the beings relative to the other, or more exactly the dependence of a transcendental degree of appearance relative to another. The dependence of the degree q in regards to the degree p is denoted p => q.

Proposition 11: Given a world and a definite appearing of this world—given, consequently, with its degree of appearance—there exists always another appearing which degree of appearance is the largest of all those which, as for their appearance, have nothing in common with the first (or, of which the conjunction with the first is equal to the minimum). This being-there, which is in regard to the first like a maximum strangeness in the world, is called its reverse. Given a degree of appearance p, the degree which measures the appearance of a being in position of towards for all beings of degree of appearance p is denoted ¬ p.

Proposition 12: The conjunction of a degree and of its reverse is always equal to the minimum. And the reverse of the reverse of a degree is always superior or equal to this degree itself. And this is denoted for the first property, p ∩ ¬ p = μ; and for the second, p ≤ ¬ ¬ p .

Proposition 13: There exists, in the transcendental of any world, a maximal degree of appearance. This maximal degree is the reverse of the degree minimal. We denote this M, and we have M = ¬ μ.

Proposition 14: The reverse of the reverse of the minimal degree is equal to this same degree. Thus . And likewise, the reverse of the reverse of the maximal degree is equal to the maximal degree. Thus ¬ ¬ M = M. In these particular cases, the double negation is equivalent to affirmation. The minimum and the maximum conduct each other, as for the double negation, in a classic way (*).

Proposition 15: Logic, in the usual sense, be it the formal calculation of the propositions and of the predicates, receives, for a given world, its values of truth and the signification of its operators from the sole transcendental of this world. In this way, formal logic is a simple consequence of transcendental logic.

Proposition 16: The world of ontology, i.e. mathematics, historically constituted, of the pure multiple, is a classic world (*).

Proposition 17: The transcendental degree which measures, in a given world, the identity of one appearing to another, measures as well the identity of this other to the first: the function of transcendental indexation is symmetrical.

Proposition 18: The intensity of co-appearance, or conjunction, in a given world of the identity of an appearing and of another, then of this other and of a third, cannot surpass the degree of identity assessable directly between the first and the third. The transcendental indexation obeys, in view of the conjunction, the triangular inequality.

Proposition 19: An appearing in a world can exist only inasmuch as it is identical to another appearing.

Proposition 20: If an element of an object (*) inexists (*) in a world, it is only minimally identical to another element of the same object.

Proposition 21: Le be a world and an appearing of this world. Let be a fixed element of the multiple which makes the Being of this appearing. The function which assigns, to every element of this multiple, the transcendental degree of its identity to the fixed element, is an atom (*) of appearance. This atom is called the real atom prescribed by the fixed element.

Proposition 22 - Postulate of materialism: Whichever the world, every atom of this world is real.

Proposition 23: Every localization (*) of an atom over a transcendental degree is also an atom.

Proposition 24: The atoms of appearance prescribed by two ontologically distinct elements of an object are nevertheless identical if, and only if, the degree of transcendental identity of these two elements is equal to their degree of existence (which is therefore the same for both). Or: if and only if they exist exactly as they are identical.

Proposition 25: Two elements of an object are compatible (*) if and only if their degree of identity is equal to the conjunction (or intersection) of their existences.

Proposition 26: If one identifies the elements of the multiple inherent to an object whose atoms they prescribe, there exists over every object a relation of order, called onto-logical, denoted as ∠ susceptible to three equivalent definitions:
—Algebraic: two elements are compatible, and the existence of the first is inferior or equal to that of the second.
—Transcendental: The existence of the first element is equal to its degree of transcendental identity to the second.
—Topological: The first element is equal to the localization of the second over the existence of the first.

Proposition 27: Let B be an objective region (*). If the elements of this region are compatible in pairs, there exists, for the relation of the onto-logical order of Proposition 26, an envelope of B, and therefore a real synthesis of this objective region.

Proposition 28: Fundamental theorem of atomic logic. Let A be a set which ontologically underlies an object (*) (A,Id) in a world m of which the transcendental is T. We denote F A, and we call the transcendental functor of A the assignment to every element p of T (or transcendental degree) of the subset of A composed of all the elements of A of which the degree of existence is p, it follows F A (p) = ⎨x / x ∈ E A and E x = p⎬. We call this territory of p, and we write Θ, every subset of T of which p is the envelope, this being p = Σ Θ. We call this at last coherent projective representation of Θ the association to every element q of Θ, of an element of F A(q), let it be Xq (manifestly we have Exq = q) which possesses the following property: for q ∈ Θ and q’ ∈ Θ the elements of F A(q) and F A(q’) corresponding, xq and xq’ are compatibles (*) between them, it follows xq ‡ xq’. Under these conditions, there always exists one and only one element e of F A(p) — p being the envelope of Θ — which is such that, for every q ∈ Θ, the localization (*) of ε over q is uniformly equal to the element xq of the coherent representation, that is ε ∫ q = xq. This element ε is the real synthesis of the subset constituted by the xq, in the sense that it is their envelope for the relation of onto-logical order noted ∠.

Proposition 29: Death (*) is a category of the logic — of appearance — and not a category of the ontology — of Being.

Proposition 30: The opens (*) of a topological space (*) have the structure of a transcendental. And the points (*) of a transcendental can be canonically endowed by a structure of plain topological space.

Proposition 31: If a transcendental has enough points (*), it is identical (isomorphic) to a plain topological space.

Proposition 32: Ontologically, the dimension of any world, measured by the number of beings which appear there, is that of an inaccessible cardinal (*). Every world is thus enclosed, but this enclosure is, from the interior of the world, inaccessible by operations of any kind.

Proposition 33: A relation between objects (*) defined as a function between the support-sets of two objects implicated in the relation, for as much as this function creates neither existence nor difference—it preserves the degree of existence of an element and never lessens the degree of identity of the two elements—, is able to preserve the set of atomic logic, notably the localizations (*), the compatibilities (*) and the onto-logical order (cf. Proposition 26). Which writes, if the elements concerned are (A,α) and (B,β), and if the relation is ρ, for a ∈ A and b ∈ A:

- Eρ(a) = Ea conservation of existence
- α(a b) ≤ β [ρ(a), ρ(β)] no creation of difference
- (a ‡ b) → [ρ(a) ‡ ρ(b)] conservation of compatibility
- (a ∠ b) → [ρ(a) ∠ ρ(b)] conservation of the onotological order

Proposition 34 - Second Thesis Constitutive of Materialism (for the first one, see the Proposition 22): It is of that alone that every world is ontologically taken in an inaccessible enclosure, inferring that every world is logically complete (*). This reflects as well: the ontological enclosure of Worlds entails their logical completeness. Or, more technically: of that which the cardinality of a world is an inaccessible infinite, deducing that every relation is universally exposed (*).

Proposition 35: Every object (*) of a world admits one—and one only—ontologically real element of which the transcendental degree of existence in this world is minimal. Or again: every object with appears in a world admits an element which inexists (*) in this world. We call this element the inexistent proper of the considered object. If (A,α) is the object, the inexistent proper is denoted ∅A.


Atom (of appearance): Let be a world of which the transdental is T, and an object (*) of this world, denoted (A,Id). We call “atom” a function of the set A towards the set (ordained) T which is such that an element added to A takes, in T, the maximal value M. In other words, an atom is a “component” of the object reduced to a known element (one of maximal appertaining value) at most. It is therefore the instance of the One in the object.

Classic (World): A classic world is a world of which the transcendental operations (conjunction and envelope) define a Boolean algebra. This means (cf. Proposition 15) that interpreted in such a world, the logic is classical. Or rather, that one has the equality of the reverse of the reverse of a degree and of this degree (law of double negation: ¬ ¬ p = p), and that the union of a degree and its reverse is equal to the maximum (law of excluded third p ∪ ¬ p = M).

Compatibility, Compatible: Let an object (*) be of any world, denoted (A,Id), and let two elements a and b be of the support-set of this object, that is to say of A. We say that a and b are compatible, which is denoted a ‡ b if (this is the most simple form of definition, but not the most “original”) the conjunction of their degrees of existence (*) is equal to their degree of identity (cf. the entry “transcendental indexation”). In other words, the “commonality” of their existences is the same thing as the measure of their identity. This is denoted Ea ∩ Eb = Id(a,b).

Completeness (Logical), Logically Complete World: A world is logically complete if every relation (cf. Proposition 33) is universally exposed (*). This property defines then the logical completeness of a world. Proposition 34, or the second constitutive thesis of materialism, says that every world is logically complete (because it is ontologically enclosed).

Death: We call “death” for an appearing in a particular world, the passage of a value of positive existence (as feeble as it can be) to the minimal value, and therefore the passage to inexisting.
“Death” designates the transition (Ex = p) ⊃ (Ex = μ). Taking into account the definition of existence (degree of identity to itself), we can define as well the death of a singular appearing, in a determined world as the coming of a complete non-identity to itself.

Exist, Existence: The degree of existence of a being is the transcendental indexation (*) of its identity to itself. We also call this degree “the existence” of the being considered (relative to its appearance in a world). The existence—as death (*)—is thus a category of appearance, and not of Being. Formally, let (A,Id) an object be in a world and let a be an element of A. The existence of a is the value in the transcendental T of Id(a,a). We denote in general Ea as the existence of a.

Exposition (of a Relation), Exposed Relation: A relation between two objects (*) is exposed (in the world) if there exists a third object which
—is itself in relation with the first two objects
—of such a fashion that the “relational triangle” commutes.
In other words, if A, B are two objects (we’re simplifying the notations), we say that r is exposed if there exists an object C, a relation f of C to A, and a relation g of C to B, relations such that the composition of r and of f are equivalent to g. Or rather: if one goes from C to B by way of A, thus enchaining f then r, this is exactly the same thing than if one goes directly from C to B by way of g. We say, then, that C is an exponent of the relation ρ.

Function of Appearance, or Function of Identity: See “transcendental indexation.”

Inaccessible Cardinal: We know that a cardinal number is the measure of the absolute number of elements of any multiple. Thus the cardinal number we write “5” measures the quantity of elements of every finite multiplicity which has 5 elements. Cantor defined infinite cardinals—we will not discuss his approach here. He also defined the order of the infinite cardinals, at the price of the admission of the axiom of choice. Among the infinite cardinals, will be said “inaccessible” those which cannot be obtained, from a smaller cardinal, through none of the two fundamental constructions of the set theory: the union which allows to pass from A to ∪ A in considering all the elements of the elements of A (dissemination), and the taking of the parts which allows to pass from A to PA in considering all the parts of A (totalization). We can say that an inaccessible cardinal is internally enclosed for the operations of dissemination and totalization: if we operate on a cardinal smaller than that to which these operations are applied, we obtain a cardinal always smaller than it. We will note that the infinite cardinal ℵ0 which is the smaller of the infinite cardinals, is nonetheless inaccessible (for the operations ∪ and P applied to the finite cardinals, give as might be expected finite cardinals). However, an inaccessible cardinal bigger than ℵ0 is absolutely gigantic, and its existence is indemonstrable: it is necessary to prescribe it by a special axiom.

Inexist, Inexistence: Given an object (A,Id) in a world, an element a of A inexists in this world if its degree of existence (*) is minimal. In other words, a inexists if Ea = μ.

Localization (on a Transcendental Degree): Proposition 21 shows that every element a of the support set A of an object (*) (A,Id) defines an atom (*) (the real atom does not prescribe a). We call localization of this atom on a transcendental degree the function which associates to every element of A the conjunction of its value for the real atom in question and of the transcendental degree. Proposition 22 shows that this conjunction (therefore the localization) is still an atom.
Formally, the real atom prescribed by the element a corresponds to every x of A the transcendental value Id(a,x), which measures the degree of identity of the Being-there of x to that of a. The localization on the degree p of this real atom makes localization to correspond to x the transcendental value Id(a,x) ∩ p.
In the atomic logic (*), we identify in general an element the atom prescribed by an element of the support-set of an object of the world to this element itself. We denote, then the localization of the prescribed atom by a over the degree p: a ∫ p. It is necessary however to remember that a ∫ p is a function (atomic) which is nothing other than the function Id(a,x) ∩ p.

Object: We call “object”, for a determined world of which the transcendental is T, the conjoint element of the set (called “support-set” of the object) and of a transcendental indexation (*) of this object over T. This is the reason for which we denote (A,Id), or again (A,α), or (B,β) etc. We say as well that an object (A, Id) is a form of the Being-there of the multiple A (in the considered world). The object is thus a category of appearance (or of logic), and not a category of the Being (or of ontology).

Objective Region: Given an object (A,Id), we call “objective region” every subset B of A.

Opens (of a Topological Space): Given a topological space (*), an open is a part of the set of base which is identical to its interior. Formally, “A is open” means that Int(A) = A. The extreme importance for us of this notion is that the opens of a topological space, ordained by inclusion (in the current sense of the set-ist inclusion), form a transcendental (that is, we find here a minimal value, the conjunction and the envelope. Cf. Propositions 5 to 9). In fact, most of the worlds have, as transcendental, the opens of a topological space. And this also causes that the majority of the worlds are not classical worlds.

Plain Topological Space: We say that a topological space (*) is plain if it has as many points (*) as its set of base E has elements.

Points (of a Transcendental): Given a transcendental T, we call “point” of the transcendental a function of T towards the subset ⎨μ,M⎬, made up of the minimum and of the maximum, as far as this function preserves the transcendental operations (is a homorphism of these operations, between T and the sub-transcendental that constitutes the pair ⎨μ,M⎬. This means that if f is the function (the point), we have:

- φ (p ∩ q) = φ (p) ∩′ φ (q)
- φ (ΣB) = Σ′⎨φ (p) / p ∈ B⎬

The sense of the notion of point is clarified if one considers that it “filters” the nuances of the transcendental (the infinite possible of the degrees) by the decisional brutality of “this, or that” which represents the simple couple of the minimum and the maximum, from zero and from one. This couple is the most classic transcendental there exists, as says Proposition 14. It is the one which interprets the usual logic, notably that of the ordinary mathematics of sets (an element belongs to a set E, or it doesn’t belong to it, there is no other transcendental possibility). A point is a global correlation, respectful of the operations between a complex transcendental (for example the opens of a topology) and the base classic transcendental which supports binary logic.
We say that a transcendental has “as enough points” if — in general — it has as many points as elements. It is in this case isomorphic to a plain topological space (*). It is necessary to know that there exist transcendentals which have no point.

Topological Space: We call topological space the conjoint element of a set E and of a function Int (said “interior of”). The function “interior” associates, to every part A of E, another part (said “interior of A”) which obeys the four fundamental axioms: the interior of A is included in A, the interior of the interior of A is nothing other than the interior of A, the interior of E is E itself, and finally the interior of the intersection of two parts A and B is precisely the intersection of their interiors.

Transcendental Indexation: The transcendental indexation of a being (of any multiplicity), relative to a given world, is a function which, to a pair of elements of the considered being, matches a transcendental degree. We say that this degree measures the identity of two elements in the world where they appear. Formally, let A be the set “supposed to appear” in a world. It appears only inasmuch as a transcendental indexation Id brings it to the transcendental T of the world in the following fashion: for every pair of elements a and b of A, we have Id(a,b) = p, where p is an element of T. We will say that a and b are, in the world in question, “identical to the degree p”. For example, if p is the minimum m of T (cf. Proposition 6), a and b are “as little identical as possible”. This means that the Being-there of a is—in this world—absolutely different than that of b.
We call as well the function Id function of appearance, or function of identity, for obvious reasons.

Universal Exposition (of a Relation): Let an exposed relation be in a world. We say that it is universally exposed if there exists an exponent such that, for every other exponent, there exists, from the second to the first, a unique relation which makes every relational triangle commute.
Formally, if A and B are the objects implicated in the relation , ( "goes" from A to B), if U is the "universal" exponent, and C is another exponent, all of the triangles commute in the diagram below:

We will say that U is the universal exponent of

L'Être et l'évenement: Tome 2, Logiques des Mondes, Paris: Broché, 2006. A forthcoming translation by Alberto Toscano as Logics of Worlds (Continuum) is due in 2008.

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