Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either. In constructive logic, a statement is 'only true' if there is a constructive proof that it is true, and 'only false' if there is a constructive proof that it is false. Operations in constructive logic preserve justification, rather than truth.

Syntactically, intuitionistic logic is a restriction of classical logic in which the law of excluded middle and double negation elimination are not axioms of the system, and cannot be proved. There are several semantics commonly employed. One semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models.

Constructive logic is practically useful because its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that given a constructive proof that an object exists, then that constructive proof can be turned into an algorithm for generating an example of it.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism.

Syntax[link]

The Rieger–Nishimura lattice. Its nodes are the propositional formulas in one variable up to intuitionistic logical equivalence, ordered by intuitionistic logical implication.

The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic. However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use →, ∧, ∨, ⊥ as the basic connectives, treating ¬A as an abbreviation for (A → ⊥). In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.

Many tautologies of classical logic can no longer be proven within intuitionistic logic. Examples include not only the law of excluded middle p ∨ ¬p, but also Peirce's law ((p → q) → p) → p, and even double negation elimination. In classical logic, both p → ¬¬p and also ¬¬p → p are theorems. In intuitionistic logic, only the former is a theorem: double negation can be introduced, but it cannot be eliminated. Rejecting p ∨ ¬p may seem strange to those more familiar with classical logic, but proving this statement in constructive logic would require producing a proof for the truth or falsity of all possible statements, which is impossible for a variety of reasons.

Because many classically valid tautologies are not theorems of intuitionistic logic, but all theorems of intuitionistic logic are valid classically, intuitionistic logic can be viewed as a weakening of classical logic, albeit one with many useful properties.

Sequent calculus[link]

Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system which is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula in this position.

Other derivatives of LK are limited to intuitionisitic derivations but still allow multiple conclusions in a sequent. LJ' ^[1] is one example.

Hilbert-style calculus[link]

Intuitionistic logic can be defined using the following Hilbert-style calculus. Compare with the deduction system at Propositional calculus#Alternative calculus.

In propositional logic, the inference rule is modus ponens

• MP: from Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to \psi
infer Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

and the axioms are

• THEN-1: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to (\chi \to \phi )

• THEN-2: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to (\chi \to \psi )) \to ((\phi \to \chi ) \to (\phi \to \psi ))

• AND-1: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \land \chi \to \phi

• AND-2: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \land \chi \to \chi

• AND-3: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to (\chi \to (\phi \land \chi ))

• OR-1: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to \phi \lor \chi

• OR-2: Failed to parse (Missing texvc executable; please see math/README to configure.): \chi \to \phi \lor \chi

• OR-3: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \psi ) \to ((\chi \to \psi ) \to (\phi \lor \chi \to \psi ))

• FALSE: Failed to parse (Missing texvc executable; please see math/README to configure.): \bot \to \phi

To make this a system of first-order predicate logic, the generalization rules

• Failed to parse (Missing texvc executable; please see math/README to configure.): \forall

-GEN: from Failed to parse (Missing texvc executable; please see math/README to configure.): \psi \to \phi

infer Failed to parse (Missing texvc executable; please see math/README to configure.): \psi \to (\forall x \ \phi )

, if Failed to parse (Missing texvc executable; please see math/README to configure.): x

is not free in Failed to parse (Missing texvc executable; please see math/README to configure.): \psi 

• Failed to parse (Missing texvc executable; please see math/README to configure.): \exists

-GEN: from Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to \psi

infer Failed to parse (Missing texvc executable; please see math/README to configure.): (\exists x \ \phi ) \to \psi 

, if Failed to parse (Missing texvc executable; please see math/README to configure.): x

is not free in Failed to parse (Missing texvc executable; please see math/README to configure.): \psi 

are added, along with the axioms

• PRED-1: Failed to parse (Missing texvc executable; please see math/README to configure.): (\forall x \ \phi (x)) \to \phi (t)

, if the term t is free for substitution for the variable x in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

(i.e., if no occurrence of any variable in t becomes bound in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi (t)

)

• PRED-2: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi (t) \to (\exists x \ \phi (x))

, with the same restriction as for PRED-1

Optional connectives[link]

Negation[link]

If one wishes to include a connective Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot

for negation rather than consider it an abbreviation for Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to \bot 

, it is enough to add:

• NOT-1': Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \bot ) \to \lnot \phi

• NOT-2': Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot \phi \to (\phi \to \bot )

There are a number of alternatives available if one wishes to omit the connective Failed to parse (Missing texvc executable; please see math/README to configure.): \bot

(false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms

• NOT-1: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \chi ) \to ((\phi \to \lnot \chi ) \to \lnot \phi )

• NOT-2: Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \to (\lnot \phi \to \chi )

as at Propositional calculus#Axioms. Alternatives to NOT-1 are Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \lnot \chi ) \to (\chi \to \lnot \phi )

or Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \lnot \phi ) \to \lnot \phi 

.

Equivalence[link]

The connective Failed to parse (Missing texvc executable; please see math/README to configure.): \leftrightarrow

for equivalence may be treated as an abbreviation, with Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \leftrightarrow \chi 
standing for Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \chi ) \land (\chi \to \phi )

. Alternatively, one may add the axioms

• IFF-1: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \leftrightarrow \chi ) \to (\phi \to \chi )

• IFF-2: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \leftrightarrow \chi ) \to (\chi \to \phi )

• IFF-3: Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \chi ) \to ((\chi \to \phi ) \to (\phi \leftrightarrow \chi ))

IFF-1 and IFF-2 can, if desired, be combined into a single axiom Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \leftrightarrow \chi ) \to ((\phi \to \chi ) \land (\chi \to \phi ))

using conjunction.

Relation to classical logic[link]

The system of classical logic is obtained by adding any one of the following axioms:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \lor \lnot \phi

(Law of the excluded middle. May also be formulated as Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \chi ) \to ((\lnot \phi \to \chi ) \to \chi )

.)

• Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot \lnot \phi \to \phi

(Double negation elimination)

• Failed to parse (Missing texvc executable; please see math/README to configure.): ((\phi \to \chi ) \to \phi ) \to \phi

(Peirce's law)

In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame Failed to parse (Missing texvc executable; please see math/README to configure.): \circ{\longrightarrow}\circ

(in other words, that is not included in Smetanich's logic).

Another relationship is given by the Gödel–Gentzen negative translation, which provides an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel–Gentzen translation is provable intuitionistically. Therefore intuitionistic logic can instead be seen as a means of extending classical logic with constructive semantics.

In 1932, Kurt Gödel defined a system of Gödel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics.

Relation to many-valued logic[link]

Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely-many valued logic. (See the section titled Heyting algebra semantics below for a sort of "infinitely-many valued logic" interpretation of intuitionistic logic.)

Non-interdefinability of operators[link]

In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together with negation, such as in Łukasiewicz's three axioms of propositional logic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation.

These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean functions. The law of bivalence does not hold in intuitionistic logic, only the law of non-contradiction. As a result none of the basic connectives can be dispensed with, and the above axioms are all necessary. Most of the classical identities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions. They are as follows:

Conjunction versus disjunction:

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \psi) \to \neg (\neg \phi \vee \neg \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \vee \psi) \to \neg (\neg \phi \wedge \neg \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\neg \phi \vee \neg \psi) \to \neg (\phi \wedge \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\neg \phi \wedge \neg \psi) \leftrightarrow \neg (\phi \vee \psi)

Conjunction versus implication:

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \psi) \to \neg (\phi \to \neg \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \psi) \to \neg (\phi \wedge \neg \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \neg \psi) \to \neg (\phi \to \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \to \neg \psi) \leftrightarrow \neg (\phi \wedge \psi)

Disjunction versus implication:

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \vee \psi) \to (\neg \phi \to \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\neg \phi \vee \psi) \to (\phi \to \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): \neg (\phi \to \psi) \to \neg (\neg \phi \vee \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): \neg (\phi \vee \psi) \leftrightarrow \neg (\neg \phi \to \psi)

Universal versus existential quantification:

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\forall x \ \phi(x)) \to \neg (\exist x \ \neg \phi(x))

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\exist x \ \phi(x)) \to \neg (\forall x \ \neg \phi(x))

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\exist x \ \neg \phi(x)) \to \neg (\forall x \ \phi(x))

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\forall x \ \neg \phi(x)) \leftrightarrow \neg (\exist x \ \phi(x))

So, for example, "a or b" is a stronger statement than "if not a, then b", whereas these are classically interchangeable. On the other hand, "not (a or b)" is equivalent to "not a, and also not b".

If we include equivalence in the list of connectives, some of the connectives become definable from others:

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi\leftrightarrow \psi) \leftrightarrow ((\phi \to \psi)\land(\psi\to\phi))

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi\to\psi) \leftrightarrow ((\phi\lor\psi) \leftrightarrow \psi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi\to\psi) \leftrightarrow ((\phi\land\psi) \leftrightarrow \phi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi\land\psi) \leftrightarrow ((\phi\to\psi)\leftrightarrow\phi)

• Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi\land\psi) \leftrightarrow (((\phi\lor\psi)\leftrightarrow\psi)\leftrightarrow\phi)

In particular, {∨, ↔, ⊥} and {∨, ↔, ¬} are complete bases of intuitionistic connectives.

As shown by Alexander Kuznetsov, either of the following connectives – the first one ternary, the second one quinary – is by itself functionally complete: either one can serve the role of a sole sufficient operator for intuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic:^[2]

• Failed to parse (Missing texvc executable; please see math/README to configure.): ((p\lor q)\land\neg r)\lor(\neg p\land(q\leftrightarrow r)),

• Failed to parse (Missing texvc executable; please see math/README to configure.): p\to(q\land\neg r\land(s\lor t)).

Semantics[link]

The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics.

Heyting algebra semantics[link]

In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid sentence of classical logic if and only if its value is 1 for every valuation—that is, for any assignment of values to its variables.

A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.

It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R.^[3] In this algebra, the ∧ and ∨ operations correspond to set intersection and union, and the value assigned to a formula A → B is int(A^C ∪ B), the interior of the union of the value of B and the complement of the value of A. The bottom element is the empty set ∅, and the top element is the entire line R. The negation ¬A of a formula A is (as usual) defined to be A → ∅. The value of ¬A then reduces to int(A^C), the interior of the complement of the value of A, also known as the exterior of A. With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line.^[3]

For example, the formula ¬(A ∧ ¬A) is valid, because no matter what set X is chosen as the value of the formula A, the value of ¬(A ∧ ¬A) can be shown to be the entire line:

Value(¬(A ∧ ¬A)) =

int((Value(A ∧ ¬A))^C) =

int((Value(A) ∩ Value(¬A))^C) =

int((X ∩ int((Value(A))^C))^C) =

int((X ∩ int(X^C))^C)

A theorem of topology tells us that int(X^C) is a subset of X^C, so the intersection is empty, leaving:

int(∅^C) = int(R) = R

So the valuation of this formula is true, and indeed the formula is valid.

But the law of the excluded middle, A ∨ ¬A, can be shown to be invalid by letting the value of A be {y : y > 0 }. Then the value of ¬A is the interior of {y : y ≤ 0 }, which is {y : y < 0 }, and the value of the formula is the union of {y : y > 0 } and {y : y < 0 }, which is {y : y ≠ 0 }, not the entire line.

The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula.^[3] Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element.^[4][5] No finite Heyting algebra has both these properties.^[3]

Kripke semantics[link]

Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics.^[6]

Relation to other logics[link]

Intutionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic or dual-intuitionistic logic.^[7]

The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic.

See also[link]

Logic portal

Notes[link]

References[link]

• Van Dalen, Dirk, 2001, "Intuitionistic Logic", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Morten H. Sørensen, Paweł Urzyczyn, 2006, Lectures on the Curry-Howard Isomorphism (chapter 2: "Intuitionistic Logic"). Studies in Logic and the Foundations of Mathematics vol. 149, Elsevier.
• W. A. Carnielli (with A. B.M. Brunner)."Anti-intuitionism and paraconsistency". Journal of Applied Logic Volume 3, Issue 1, March 2005, pages 161-184.

External links[link]

• Stanford Encyclopedia of Philosophy: "Intuitionistic Logic" -- by Joan Moschovakis.

Alain Badiou

Alain Badiou. Fnac Montparnasse (Paris). February 5th, 2010
Born	(1937-01-17) 17 January 1937 (age 75) Rabat, French Morocco
Era	Contemporary philosophy
Region	French philosophy
School	Marxism Continental philosophy
Main interests	Set Theory, Mathematics, Metapolitics, Ontology, Marxism
Notable ideas	Événement (Event), ontologie du multiple (ontology of the multiple & ontology is mathematics), Un n'est pas ("The One is Not")
Influenced by Plato, Marx, Cantor, Albert Lautman, Mao Zedong, Lacan, Althusser, Paul Cohen, Sartre, Deleuze, Lyotard, Hegel, Mallarmé, Beckett, Fernando Pessoa, Sylvain Lazarus
Influenced Slavoj Žižek, Peter Hallward, Simon Critchley, Ray Brassier, Sylvain Lazarus, Jason Barker, Quentin Meillassoux, Bruno Bosteels, Alberto Toscano, Oliver Feltham, Justin Clemens, Nina Power

Alain Badiou (French pronunciation: [alɛ̃ badju]  (listen) (help·info); born 17 January 1937 in Rabat, Morocco) is a French philosopher, professor at European Graduate School, formerly chair of Philosophy at the École Normale Supérieure (ENS).^[1]

Badiou has written about the concepts of being, truth and the subject in a way that, he claims, is neither postmodern nor simply a repetition of modernity. Politically, Badiou is committed to the far left, and to the communist tradition.

Slavoj Žižek has written of Badiou that he is "a figure like Plato or Hegel walk[ing] here among us".^[2]

Biography[link]

Badiou was a student at the Lycée Louis-Le-Grand and then the Ecole Normale Supérieure (1957–1961). He taught at the lycée in Reims from 1963 where he became a close friend of fellow playwright (and philosopher) François Regnault,^[3] and published a couple of novels before moving to the University of Paris VIII (Vincennes-Saint Denis) in 1969.^[4] Badiou was politically active very early on, and was one of the founding members of the Unified Socialist Party (PSU). The PSU was particularly active in the struggle for the decolonization of Algeria. He wrote his first novel, Almagestes, in 1964. In 1967 he joined a study group organized by Louis Althusser, became increasingly influenced by Jacques Lacan and became a member of the editorial board of Cahiers pour l’Analyse.^[4] By then he "already had a solid grounding in mathematics and logic (along with Lacanian theory)",^[4] and his own two contributions to the pages of Cahiers "anticipate many of the distinctive concerns of his later philosophy".^[4]

The student uprisings of May 1968 reinforced Badiou's commitment to the far Left, and he participated in increasingly militant groups, such as the Union des communistes de France marxiste-léniniste (UCFml). To quote Badiou himself, the UCFml is "the Maoist organization established in late 1969 by Natacha Michel, Sylvain Lazarus, myself and a fair number of young people".^[5] During this time, Badiou joined the faculty of the newly founded University of Paris VIII/Vincennes-Saint Denis which was a bastion of counter-cultural thought. There he engaged in fierce intellectual debates with fellow professors Gilles Deleuze and Jean-François Lyotard, whose philosophical works he considered unhealthy deviations from the Althusserian program of a scientific Marxism.

In the 1980s, as both Althusserian Marxism and Lacanian psychoanalysis went into decline (with Lacan dead and Althusser in an asylum), Badiou published more technical and abstract philosophical works, such as Théorie du sujet (1982), and his magnum opus, Being and Event (1988). Nonetheless, Badiou has never renounced Althusser or Lacan, and sympathetic references to Marxism and psychoanalysis are not uncommon in his more recent works (most notably Petit panthéon portatif/Pocket Pantheon).^[6][7]

He took up his current position at the ENS in 1999. He is also associated with a number of other institutions, such as the Collège International de Philosophie. He was a member of "L'Organisation Politique" which, as mentioned above, he founded in 1985 with some comrades from the Maoist UCFml. This organization disbanded in 2007, according to the French Wikipedia article (linked to in the previous sentence). Badiou has also enjoyed success as a dramatist with plays such as Ahmed le Subtil.

In the last decade, an increasing number of Badiou's works have been translated into English, such as Ethics, Deleuze, Manifesto for Philosophy, Metapolitics, and Being and Event. Short pieces by Badiou have likewise appeared in American and English periodicals, such as Lacanian Ink, New Left Review, Radical Philosophy, Cosmos and History [1] and Parrhesia. Unusually for a contemporary European philosopher his work is increasingly being taken up by militants in movements of the poor in countries like India, the Democratic Republic of Congo and South Africa where he is often read together with Frantz Fanon.^{[citation needed]}

Lately Badiou got into a fierce controversy within the confines of Parisian intellectual life. It started in 2005 with the publication of his "Circonstances 3: Portées du mot 'juif'" – The Uses of the Word "Jew".^[8] This book generated a strong response with calls of Badiou being labelled Anti-Semitic. The wrangling became a cause célèbre with articles going back and forth in the French newspaper Le Monde and in the cultural journal "Les temps modernes." Another philosopher, Jean-Claude Milner, has accused Badiou of Anti-Semitism.^[9]

Key concepts[link]

Badiou makes repeated use of several concepts throughout his philosophy. One of the aims of his thought is to show that his categories of truth are useful for any type of philosophical critique. Therefore, he uses them to interrogate art and history as well as ontology and scientific discovery. Johannes Thumfart argues that Badiou's philosophy can be regarded as a contemporary reinterpretation of Platonism.^[10]

Conditions[link]

According to Badiou, philosophy is suspended from four conditions (art, love, politics, and science), each of them fully independent "truth procedures." (For Badiou's notion of truth procedures, see below.) Badiou consistently maintains throughout his work (but most systematically in Manifesto for Philosophy) that philosophy must avoid the temptation to suture itself (that is, to hand over its entire intellectual effort) to any of these independent truth procedures. When philosophy does suture itself to one of its conditions (and Badiou argues that the history of philosophy during the nineteenth and twentieth centuries is primarily a history of sutures), what results is a philosophical "disaster." Consequently, philosophy is, according to Badiou, a thinking of the compossibility of the several truth procedures, whether this is undertaken through the investigation of the intersections between distinct truth procedures (the intersection of art and love in the novel, for instance), or whether this is undertaken through the more traditionally philosophical work of addressing categories like truth or the subject (concepts that are, as concepts, external to the individual truth procedures, though they are functionally operative in the truth procedures themselves). For Badiou, when philosophy addresses the four truth procedures in a genuinely philosophical manner, rather than through a suturing abandonment of philosophy as such, it speaks of them with a theoretical terminology that marks its philosophical character: inaesthetics rather than art; metapolitics rather than politics; ontology rather than science; etc.

Truth, for Badiou, is a specifically philosophical category. While philosophy's several conditions are, on their own terms, "truth procedures" (i.e., they produce truths as they are pursued), it is only philosophy that can speak of the several truth procedures as truth procedures. (The lover, for instance, does not think of her love as a question of truth, but simply and rightly as a question of love. Only the philosopher sees in the true lover's love the unfolding of a truth.) Badiou has a very rigorous notion of truth, one that is strongly against the grain of much of contemporary European thought. Badiou at once embraces the traditional modernist notion that truths are genuinely invariant (always and everywhere the case, eternal and unchanging) and the incisively postmodernist notion that truths are constructed through processes. Badiou's theory of truth, exposited throughout his work, accomplishes this strange mixture by uncoupling invariance and self-evidence (such that invariance does not imply self-evidence), as well as by uncoupling constructedness from relativity (such that constructedness does not lead to relativism).

The idea, here, is that a truth's invariance makes it genuinely indiscernible: because a truth is everywhere and always the case, it passes unnoticed unless there is a rupture in the laws of being and appearance, during which the truth in question becomes, but only for a passing moment, discernible. Such a rupture is what Badiou calls an event, according to a theory originally worked out in Being and Event and fleshed out in important ways in Logics of Worlds. The individual who chances to witness such an event, if he is faithful to what he has glimpsed, can then introduce the truth by naming it into worldly situations. For Badiou, it is by positioning oneself to the truth of an event that a human animal becomes a subject; subjectivity is not an inherent human trait. According to a process or procedure that subsequently unfolds only if those who subject themselves to the glimpsed truth continue faithful in the work of announcing the truth in question, genuine knowledge is produced (knowledge often appears in Badiou's work under the title of the "veridical"). While such knowledge is produced in the process of being faithful to a truth event, it should be noted that, for Badiou, knowledge, in the figure of the encyclopedia, always remains fragile, subject to what may yet be produced as faithful subjects of the event produce further knowledge. According to Badiou, truth procedures proceed to infinity, such that faith (fidelity) outstrips knowledge. (Badiou, following both Lacan and Heidegger, distances truth from knowledge.) The dominating ideology of the day, which Badiou terms "democratic materialism," denies the existence of truth and only recognize "bodies" and "languages." Badiou proposes a turn towards "dialectical materialism," which recognizes that there are only bodies and languages, except there are also truths.

Inaesthetic[link]

In Handbook of Inaesthetics Badiou coins the phrase "inaesthetic" to refer to a concept of artistic creation that denies "the reflection/object relation". Reacting against the idea of mimesis, or poetic reflection of "nature", Badiou claims that art is "immanent" and "singular". It is immanent in the sense that its truth is given in its immediacy in a given work of art, and singular in that its truth is found in art and art alone. His view of the link between philosophy and art is tied into the motif of pedagogy, which he claims functions so as to "arrange the forms of knowledge in a way that some truth may come to pierce a hole in them". He develops these ideas with examples from the prose of Samuel Beckett and the poetry of Stéphane Mallarmé and Fernando Pessoa (who he argues has developed a body of work that philosophy is currently incapable of incorporating), among others.

[edit] Introduction to Being and Event

The major propositions of Badiou's philosophy all find their basis in Being and Event, in which he continues his attempt (which he began in Théorie du sujet) to reconcile a notion of the subject with ontology, and in particular post-structuralist and constructivist ontologies.^[11] A frequent criticism of post-structuralist work is that it prohibits, through its fixation on semiotics and language, any notion of a subject. Badiou's work is, by his own admission,^[12] an attempt to break out of contemporary philosophy's fixation upon language, which he sees almost as a straitjacket. This effort leads him, in Being and Event, to combine rigorous mathematical formulae with his readings of poets such as Mallarmé and Hölderlin and religious thinkers such as Pascal. His philosophy draws upon both 'analytical' and 'continental' traditions. In Badiou's own opinion, this combination places him awkwardly relative to his contemporaries, meaning that his work had been only slowly taken up.^[13] Being and Event offers an example of this slow uptake, in fact: it was translated into English only in 2005, a full seventeen years after its French publication.

As is implied in the title of the book, two elements mark the thesis of Being and Event: the place of ontology, or 'the science of being qua being' (being in itself), and the place of the event – which is seen as a rupture in being – through which the subject finds realization and reconciliation with truth. This situation of being and the rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (with the axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority of either mathematicians or philosophers.

Mathematics as ontology[link]

	Set theory portal
	Mathematics portal

For Badiou the problem which the Greek tradition of philosophy has faced and never satisfactorily dealt with is that while beings themselves are plural, and thought in terms of multiplicity, being itself is thought to be singular; that is, it is thought in terms of the one. He proposes as the solution to this impasse the following declaration: that the one is not. This is why Badiou accords set theory (the axioms of which he refers to as the Ideas of the multiple) such stature, and refers to mathematics as the very place of ontology: Only set theory allows one to conceive a 'pure doctrine of the multiple'. Set theory does not operate in terms of definite individual elements in groupings but only functions insofar as what belongs to a set is of the same relation as that set (that is, another set too). What separates sets out therefore is not an existential positive proposition, but other multiples whose properties validate its presentation; which is to say their structural relation. The structure of being thus secures the regime of the count-as-one. So if one is to think of a set – for instance, the set of people, or humanity – as counting as one the elements which belong to that set, it can then secure the multiple (the multiplicities of humans) as one consistent concept (humanity), but only in terms of what does not belong to that set. What is, in following, crucial for Badiou is that the structural form of the count-as-one, which makes multiplicities thinkable, implies that the proper name of being does not belong to an element as such (an original 'one'), but rather the void set (written Ø), the set to which nothing (not even the void set itself) belongs. It may help to understand the concept 'count-as-one' if it is associated with the concept of 'terming': a multiple is not one, but it is referred to with 'multiple': one word. To count a set as one is to mention that set. How the being of terms such as 'multiple' does not contradict the non-being of the one can be understood by considering the multiple nature of terminology: for there to be a term without there also being a system of terminology, within which the difference between terms gives context and meaning to any one term, does not coincide with what is understood by 'terminology', which is precisely difference (thus multiplicity) conditioning meaning. Since the idea of conceiving of a term without meaning does not compute, the count-as-one is a structural effect or a situational operation and not an event of truth. Multiples which are 'composed' or 'consistent' are count-effects; inconsistent multiplicity is the presentation of presentation.

Badiou's use of set theory in this manner is not just illustrative or heuristic. Badiou uses the axioms of Zermelo–Fraenkel set theory to identify the relationship of being to history, Nature, the State, and God. Most significantly this use means that (as with set theory) there is a strict prohibition on self-belonging; a set cannot contain or belong to itself. Russell's paradox famously ruled that possibility out of formal logic. (This paradox can be thought through in terms of a 'list of lists that do not contain themselves': if such a list does not write itself on the list the property is incomplete, as there will be one missing; if it does, it is no longer a list that does not contain itself.) So too does the axiom of foundation – or to give an alternative name the axiom of regularity – enact such a prohibition (cf. p. 190 in Being and Event). (This axiom states that all sets contain an element for which only the void [empty] set names what is common to both the set and its element.) Badiou's philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the 'one': there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God. Badiou is therefore – against Cantor, from whom he draws heavily – staunchly atheist. However, secondly, this prohibition prompts him to introduce the event. Because, according to Badiou, the axiom of foundation 'founds' all sets in the void, it ties all being to the historico-social situation of the multiplicities of de-centred sets – thereby effacing the positivity of subjective action, or an entirely 'new' occurrence. And whilst this is acceptable ontologically, it is unacceptable, Badiou holds, philosophically. Set theory mathematics has consequently 'pragmatically abandoned' an area which philosophy cannot. And so, Badiou argues, there is therefore only one possibility remaining: that ontology can say nothing about the event.

The event and the subject[link]

The principle of the event is where Badiou diverges from the majority of late twentieth century philosophy and social thought, and in particular the likes of Foucault, Butler, Lacan and Deleuze, among others. In short, it represents that which is outside of ontology. Badiou's problem here is, unsurprisingly, the question of how to 'make use' of that which cannot be discerned. But it is a problem he views as vital, because if one constructs the world only from that which can be discerned and therefore given a name, it results in either the destitution of subjectivity and the removal of the subject from ontology (the criticism continually leveled at Foucault's discursive universe), or the Panglossian solution of Leibniz: that God is language in its supposed completion.

Badiou again turns here to mathematics and set theory – Badiou's language of ontology – to study the possibility of an indiscernible element existing extrinsically to the situation of ontology. He employs the strategy of the mathematician Paul J. Cohen, using what are called the conditions of sets. These conditions are thought of in terms of domination, a domination being that which defines a set. (If one takes, in binary language, the set with the condition 'items marked only with ones', any item marked with zero negates the property of the set. The condition which has only ones is thus dominated by any condition which has zeros in it [cf. p. 367-71 in Being and Event].) Badiou reasons using these conditions that every discernible (nameable or constructible) set is dominated by the conditions which don't possess the property that makes it discernible as a set. (The property 'one' is always dominated by 'not one'.) These sets are, in line with constructible ontology, relative to one's being-in-the-world and one's being in language (where sets and concepts, such as the concept 'humanity', get their names). However, he continues, the dominations themselves are, whilst being relative concepts, not necessarily intrinsic to language and constructible thought; rather one can axiomatically define a domination – in the terms of mathematical ontology – as a set of conditions such that any condition outside the domination is dominated by at least one term inside the domination. One does not necessarily need to refer to constructible language to conceive of a 'set of dominations', which he refers to as the indiscernible set, or the generic set. It is therefore, he continues, possible to think beyond the strictures of the relativistic constructible universe of language, by a process Cohen calls forcing. And he concludes in following that while ontology can mark out a space for an inhabitant of the constructible situation to decide upon the indiscernible, it falls to the subject – about which the ontological situation cannot comment – to nominate this indiscernible, this generic point; and thus nominate, and give name to, the undecidable event. Badiou thereby marks out a philosophy by which to refute the apparent relativism or apoliticism in post-structuralist thought.

Badiou's ultimate ethical maxim is therefore one of: 'decide upon the undecidable'. It is to name the indiscernible, the generic set, and thus name the event that re-casts ontology in a new light. He identifies four domains in which a subject (who, it is important to note, becomes a subject through this process) can potentially witness an event: love, science, politics and art. By enacting fidelity to the event within these four domains one performs a 'generic procedure', which in its undecidability is necessarily experimental, and one potentially recasts the situation in which being takes place. Through this maintenance of fidelity, truth has the potentiality to emerge.

In line with his concept of the event, Badiou maintains, politics is not about politicians, but activism based on the present situation and the evental [sic] (his translators' neologism) rupture. So too does love have this characteristic of becoming anew. Even in science the guesswork that marks the event is prominent. He vigorously rejects the tag of 'decisionist' (the idea that once something is decided it 'becomes true'), but rather argues that the recasting of a truth comes prior to its veracity or verifiability. As he says of Galileo (p. 401):

When Galileo announced the principle of inertia, he was still separated from the truth of the new physics by all the chance encounters that are named in subjects such as Descartes or Newton. How could he, with the names he fabricated and displaced (because they were at hand – ‘movement’, ‘equal proportion’, etc.), have supposed the veracity of his principle for the situation to-come that was the establishment of modern science; that is, the supplementation of his situation with the indiscernible and unfinishable part that one has to name ‘rational physics’?

Badiou, whilst keen to stress the non-equivalence between politics and philosophy, thus finds his political approach – one of activism, militancy, and scepticism of parliamentary-democratic process – backed up by his philosophy based around singular, situated truths, and potential revolutions.

L'Organisation Politique[link]

Alain Badiou is a founding member (along with Natacha Michel and Sylvain Lazarus) of the militant French political organisation L'Organisation Politique, which was active from 1985 until it disbanded in 2007.^[14] It called itself a post-party organization concerned with direct popular intervention in a wide range of issues (including immigration, labor, and housing). In addition to numerous writings and interventions, L'Organisation Politique stressed the importance of developing political prescriptions concerning undocumented migrants (les sans papiers), stressing that they must be conceived primarily as workers and not immigrants.^[15]

Works[link]

English translations[link]

Journals[link]

• The International Journal of Badiou Studies
• "The Cultural Revolution: The Last Revolution?", transl. by Bruno Bosteels; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847
• "Selections from Théorie du sujet on the Cultural Revolution", transl. by Alberto Toscano with the assistance of Lorenzo Chiesa and Nina Power; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847
• "Further Selections from Théorie du sujet on the Cultural Revolution", transl. by Lorenzo Chiesa; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847
• “The Triumphant Restoration", transl. by Alberto Toscano; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847
• "An Essential Philosophical Thesis: 'It Is Right to Rebel against the Reactionaries'", transl. by Alberto Toscano; positions: asia critique, Volume 13, Issue 3, Winter 2005; (Durham: Duke University Press, 2005): ISSN 1067-9847

DVD[link]

Lectures[link]

Further reading[link]

Notes[link]

External links[link]

Wikimedia Commons has media related to: Alain Badiou

Wikiquote has a collection of quotations related to: Alain Badiou

• Alain Badiou Faculty Page at European Graduate School
• Alain Badiou Bibliography at Lacan Dot Com
• What is a philosophical Institution? or: Address, Transmission, Inscription. Cosmos and History: The Journal of Natural and Social Philosophy, Vol 2, No 1-2 (2006)

Critical Opinions[link]

• Alain Badiou's "Politics of Emancipation": A Communism Locked Within the Confines of the Bourgeois World by Raymond Lotta, Nayi Duniya, and K. J. A.
• On Alain Badiou and Logiques des mondes by Slavoj Zizek
• The Marxist hypothesis: a response to Alain Badiou's "communist hypothesis" by Chris Cutrone