Open Philosophy 2018; 1: 213–222 Becky Vartabedian* Negation, Structure, Transformation: Alain Badiou And The New Metaphysics https://doi.org/10.1515/opphil-2018-0015 Received June 17, 2018; accepted August 23, 2018 Abstract: In this article, I discuss Alain Badiou’s 2008 address titled “The Three Negations.” Though the text was originally presented in a symposium concerning the relationship of law to Badiou’s theory of the event, I discuss the way this brief address offers an introduction to the broad sweep of Badiou’s metaphysics, outlining his accounts of being, appearing, and transformation. To do so, Badiou calls on the resources of three paradigms of negation: from classical Aristotelian logic, from Brouwer’s intuitionist logic, and in paraconsistent logics developed by DaCosta. I explain Badiou’s use of negation in the three primary areas of his metaphysics, as well as to diagnose the degrees of transformation that may have occurred in a situation. My analysis of Badiou’s use of negation in this text is aided by examples from his broader ontological oeuvre. I also explain the underlying requirement in Badiou’s work that formal considerations – mathematical or logical – get their sense by being tethered to readily-identifiable political, aesthetic, scientific, or interpersonal concerns. I conclude by addressing the foundation Badiou’s work establishes for pursuing a new metaphysics, and by discussing certain of the liabilities that remain in the wake of his account. Keywords: Alain Badiou, Aristotle, event, intuitionist logic, paraconsistent logic, law, Newton DaCosta, ontology, Markus Gabriel, negation, analytic/continental divide. The “new metaphysics” marks an increasingly porous border between continental and analytic approaches to questions concerning being, appearing, transformation, and truth. In service of these questions and in what follows, I attend to the contents of Alain Badiou’s 2008 address titled “The Three Negations.” I discuss its status as an introduction to his broad metaphysical project, which continues to unfold across three volumes: L’Être et l’événement (1988; published in English as Being and Event in 2005); Logiques des Mondes (2006; in English as Logics of Worlds in 2009), and a forthcoming third volume titled L’immanence des vérités (The Immanence of Truths).1 Though “The Three Negations” lacks the technical specificity and formal mathematical and logical apparatuses of the full-length texts, what is perhaps remarkable about Badiou’s 2008 address is its succinct capture of concepts at the core of his metaphysics. In this presentation, Badiou mines classical Aristotelian, intuitionist, and paraconsistent logics for their respective accounts of negation. He uses these varieties of negation to support an account of structure at the level of being, appearing, and transformation (respectively), and to describe degrees or intensities of transformation as strong, weak, or false. “The Three Negations” also reminds Badiou’s reader of a deep requirement of his metaphysics: that it be legible in political, aesthetic, scientific, or amorous relationship. I briefly discuss this requirement and conclude by addressing the resources Badiou’s work offers for pursuing a new metaphysics, as well as certain of the questions and liabilities that remain after Badiou’s work. 1 Though the publication of this text was proposed for 2017, it has not (as of this writing) yet appeared. In fall 2017, Badiou offered a seminar on “The Immanence of Truths” at Princeton University. Badiou’s opening lecture (in English), available online at https://www.youtube.com/watch?v=rF9Xhj0agjs. The content of Badiou’s final seminar (2015-2017) on the topic is transcribed at http://www.entretemps.asso.fr/Badiou/15-16.htm. *Corresponding author: Becky Vartabedian, Regis University, Denver, United States of America; E-mail: [email protected] Open Access. © 2018 Becky Vartabedian, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. 214 B. Vartabedian 1 Occasion of the address “The Three Negations” is Badiou’s contribution to a symposium on his work, convened at Cardozo Law School in 2008. Titled “Law and Event”, the symposium addressed the relation of event and truth– concepts used to indicate transformation in Badiou’s work – to jurisprudence and legal theory.2 In Peter Goodrich’s prefatory remarks, he introduces Badiou’s work to the audience and explains the significance of Badiou’s work for understanding the law and its capacity for transformation. Goodrich acquaints the Cardozo audience with Badiou by situating him as “French and not-quite- French,” insofar as Badiou’s work resists the kinds of characterization that attended work by Foucault, Deleuze, and Derrida in Anglo-American circles.3 According to Goodrich, Badiou and his work are unique, traversing otherwise consolidated modes of discourse. These include “traditional” French theory, expressed in Badiou’s link to work by Lacan and Althusser; a theoretical apparatus balancing interests in subjectivity and ontology with formal systems of logic and mathematics, evident in Badiou’s early The Concept of Model (1969) through the contemporary major works; and an approach that negotiates politically-charged metaphysics in a way, speaking of Being and Event, that is not “of obvious appeal to the humanities.”4 In identifying these traversals, Goodrich concludes his remarks by noting that Badiou represents “a certain rigor of thought and for a political apprehension of literature, ethics, epistemology,” and the immediate context of Goodrich’s remarks, “tentatively, law.”5 The law is, by nature, designed to resist the irruption of the novel and dispatch contradiction using established precedent; put another way, the event is prohibited by the law from appearing. As Goodrich explains, the law, “by convention or position, always already exists. It is promulgation or precedent, prerogative or pontification, custom or use, and it is law precisely to the extent of its prior existence.”6 The law’s credibility, then, depends on its deep-seated status (on the one hand), which underwrites its ability to maintain social stability.7 However, in the wake of an event, Goodrich explains, “The legal is abandoned in favor of the non-legal, indeed the illegal.”8 It is this contradictory relationship, the pressure an event exerts on the law and its function in negating the law, which forms the focus of the Cardozo symposium. 2 “The Three Negations”: Being, appearing, and varieties of transformation “The Three Negations” addresses the prospects for transformation and its relation to structure by addressing logic and its laws, focusing especially on negation, which expresses the status of the law of non-contradiction (LNC) and the law of excluded middle (LEM) in a given logical framework. Badiou is interested in negation in in Aristotle’s work, since the principles of LNC, LEM, and double negation form 2 The contents of the symposium are available at http://www.cardozolawreview.com/volume-29-issue-5.html. 3 Goodrich, “Preface.” Goodrich cites François Cusset’s French Theory: How Foucault, Derrida, Deleuze & Co. Transformed the Intellectual Life of the United States as describing a “characteristic” reception of French theory in the U.S. As Goodrich points out, Badiou receives only “scant mention” in a couple of places in Cusset’s text, considered briefly alongside Jacques Rancière, and in a discussion of Badiou’s Ethics: An Essay on the Understanding of Evil (originally L’ethique: Essai sur la conscience du Mal). Cusset discusses Ethics in connection with Badiou’s thoroughgoing critique of the “New Philosophers” in France, a group of thinkers that disavowed their leftist commitments in the 1970s and 1980s, and a target of Badiou’s ire from 1977 to the present (Cusset, French Theory, xix). 4 Goodrich, “Preface,” 1868. 5 Ibid., 1869. 6 Ibid., 1870. 7 On this point, Goodrich explains, “the authors of common law state with some stringency that where law is vague, uncertain, ill-defined, the sociality exists in a state of misery” (“Preface,” 1870). 8 Goodrich, “Preface,” 1870. Negation, Structure, Transformation: Alain Badiou And The New Metaphysics 215 “the core of classical logic.”9 From this point of origin, Badiou follows the trajectory of negation to forms of logic beyond the classical, to intuitionist logic (particularly that of L.E. J. Brouwer) and paraconsistent logic (developed by the Brazilian logician Newton DaCosta). As I explain in the following two sections, Badiou understands this trajectory according to two valences: the first is the development of negation as a tool in three key areas of his metaphysics – being, appearing, and event; the second is the use of negation to diagnose degrees of transformation as strong, weak or false. 3 Negation in being, appearing, and the event With his focus trained on the formal expression of negation, Badiou begins by identifying four modes in which negation may be deployed. The first, following the Aristotelian paradigm, maintains both LNC and LEM: one cannot assert propositions P and non-P at the same time and in the same context (LNC); and either P is true and non-P is false, or non-P true and P false – there is no third option (LEM). Aristotle presents these laws in Metaphysics Gamma as antidotes to “the unqualified doctrine which would prevent us from determining anything in our thought,” and ensure the quality of judgments about the world.10 For Badiou, this classical paradigm organizes