“There is a sort of univocity of being, but an equivocity of existence.” – Alain Badiou, ‘The Three Negations’
In a lecture delivered at the Cardozo Law School in New York City in 2008, Alain Badiou recapitulated his understanding of Being, Event, and Simulacrum in relationship to Logic and Law. With an incredible power of precision, Badiou recapitulates Aristotle’s three main pillars of the process of thought (Identity, non-contradiction, and the excluded middle), which he then uses to delineate the three kinds of negation he understands to be at work: classical, intuitionistic, and paraconsistent. With the “three kinds of negation,” Badiou aims to underscore how Events transpire in a world; and what the impact of an Event, according to each kind of negation, actually means.
In Classical logic, negation obeys Aristotle’s principles of non-contradition and the excluded middle. That is to say, the relationship between P and non-P is that either the former is true, or the latter is true, but not both simultaneously. Additionally, there is no third term available in this truth relation. All throughout Badiou’s lecture, he provides us with examples of such a logic. For Badiou, the most common understanding of how classical logic defines a certain kind of negation is seen in the concept of God: “Certainly, God as such pertains to classical logic: between his existence and his non-existence, there is no third possibility” (TTN). Badiou makes it explicit that classical logic pertains to God, only because the concept of God itself is an ontological concept. Thus we arrive at the first of Badiou’s main themes: Being (which operates according to classical logic and it’s specific kind of negation).
In intuitionistic logic, negation obeys the law of non-contradiction but does not obey the law of the excluded middle. So according to intuitionistic logic, the relation between P and non-P does not excluded any number of intermediary possibilities between those two extremes. Now, intuitionistic logic, as one might be able to already see, cannot pertain to be Being qua being (either God exists or does not exist, and it cannot be said that God exists between those two claims). However, Badiou finds intuitionistic logic useful (“valid”) when it comes to making claims about concrete worlds. Thus, keeping in mind his audience is a room full of law students and professors, he gives an example through the institution of law as to how intuitionistic logic allows us to comprehend the world:
So, if the great field of the law is always a concrete world, or a concrete construction, its logic is not classic. If we take “law” in its strict legal sense, we know that perfectly well. If the sentence P is “guilty,” and non-P “innocent,” we have always a great number of intermediate values, like “guilty with attenuating circumstances,” or “innocent because certainly guilty, but with insufficient proof,” and so on. (TTN)
Thirdly, in paraconsistent logic, negation obeys the law of the excluded middle but not the law of non-contradiction. This is defined by Badiou as “non-perceptible change at the level of the inexistent.” However, in order to get a better understanding of how paraconsistent logic fits into the ‘three kinds of negation’ Badiou is articulating, it’s helpful to turn to his own example which he believes spells this out more clearly. Regarding Events occurring in a world, Badiou claims that we have the twofold task of defining the event ontologically (abiding by the rules of classical logic) and existentially (abiding by the rules of intuitionistic logic):
To be complete, we must define first an event at the ontological level: what sort of multiplicity is an event? And after that, we must define an event at the phenomenological or existential level: how does an event appear in a determinate world? Today, and for you, I simplify the matter. I suppose that an event is a sudden change of the rules of appearing; a change of the degrees of existence of a lot of multiplicities which appear in a world…For example, the political existence of poor workers in a revolutionary event…The question for an event is: what is the destiny, after the event, of an inexistent of the world? What becomes of the poor worker after the revolution? (TTN)
In order to account for the nature of an Event in a world, Badiou redeploys the three kinds of logic in order to trace an Event in its most effective form (classical) to an Event in its least effective form, that is to say as a non-Event (paraconsistent). If the most effective Events are those which operate under classical logic it is because the Event, as that which institutes a disruption of the reason or conventions of a world, brings about the strongest contrasts between existents and inexistents in a world. As Badiou writes,
“The test is that among the consequences of this change, we have the maximal value, the maximal intensity of existence, for an object which was an inexistent, which appeared with the minimal degree of intensity. The poor worker, who before the revolution appears as nothing in the political field, becomes the new hero of this field. The abstract painting, which was purely decorative before an artistic revolution, becomes an essential trend of the history of the arts, and so on” (TTN).
Thus, for Badiou, the Event which is most effective in disrupting the conventions and rationale of a world is that which can reduce the world into a duality between minimal intensity, or inexistence, and maximal intensity. “And that sort of world, with only two degrees of intensity, is always classical. We shall say in this case that the change is a true event, simply, if the context is clear, an Event” (TTN). Now, the Event which accords to intuitionistic logic is the second possibility of an Event’s occurrence in the world. This type of Event institutes neither maximal nor minimal change, but rather intermediate changes in the world. “The poor worker appears in the political field, but it is not at all a new hero of the field. The abstract figures can be used in painting, but they are not really important. In this case, the logical framework of the event, and of its consequences, is clearly intuitionistic. There is no obligation for the event to be of maximal intensity” (TTN). This type of Event, which institutes changes in a world that do not cause fundamental breaks, shifts, or novel ways of doing art, politics, or science, abide by the principle of non-contradiction but not by the law of the excluded middle – hence, it’s intuitionistic logical nature.
Finally, we arrive at the Event which corresponds to paraconsistent logic. This kind of event is characterized as the indecidability between event and non-event. “Yes, something happens, but, from the point of view of the world, everything is identical. Se we have event and non-event simultaneously. And there are no new values between affirmation and negation, because the world is exactly the same. The principle of excluded middle is true, the principle of contradiction is false; so we have a paraconsistent logic. We say then that we have a false event, or a simulacrum“(TTN, my emphasis). Thus, for Badiou, true change only occurs in a world when the Event alters or interferes with the rules which govern a world – hence why he still claims that change occurs when Events abide by classical and intuitionistic logic (the former being a radical change, the latter being reformist). It is because of this that Badiou ends his lecture with this statement: “The lesson is that, when the world is intuitionistic, a true change must be classical, and a false change paraconsistent” (TTN).
While there is much to point out regarding Badiou’s work on these three understandings of negation and how they relate to Being, Event, and Simulacrum, I would like to make a gesture toward another thinker who highlights an important aspect of the relationship between the World and Simulacrum: Jacques Rancière.
In his text On The Shores of Politics Rancière underscores the police as the exemplary form for society to ensure ‘non-events,’ or simulacrums:
Police intervention in public space is less about interpellating demonstrators than it is about dispersing them. The police are not the law that interpellates the individual (the “hey, you there” of Louis Althusser) unless we confuse the law with religious subjection. The police are above all a certitude about what is there, or rather, about what is not there: “Move along, there’s nothing to see.” The police say there is nothing to see, nothing happening, nothing to be done but to keep moving, circulating; they say that the space of circulation is nothing but the space of circulation. Politics consists in transforming that space of circulation into the space of the manifestation of a subject: be it the people, workers, citizens. It consists in refiguring that space, what there is to do there, what there is to see, or to name. It is a dispute about the division of what is perceptible to the senses.” (OSP)
And there, in the ending of this passage, Rancière meets with Badiou, but coming from a different side: with Badiou we understand the simulacrum by its logical formation, by its being the non-event par excellence. With Rancière, we understand the non-event, the simulacrum, as being guaranteed by the police. The power of the police, according to Rancière, is to render what is sensible, existent, and manifest in public space, imperceptible, inexistent (Badiou), and silent. However, if we accept the similarities and productive relations between Badiou and Rancière, we are obliged to re-write Badiou’s summary lesson from his lecture. Now, it is not only that the world is intuitionistic, true change classical, and the false change paraconsistent. Rather, the lesson is that, when the world is paraconsistent, a true change must be classical, and a false change intuitionistic.
Thus, we might say, what is necessary for a true change in the world, is to wage a war against the world itself. This would be the underlying Heraclitean principle of a Rancièrean correlate to Badiou’s formulation. War is the father and king of all, and has produced some as gods and some as men, and has made some slaves and some free. It is the classical logic of war, with which Badiou began his lecture, that we find the potential for instituting a rupture within the conventions and rationale of a paraconsistent world.