Alain Badiou: The Three Fundamental Logics of Negation (Hyper-Transcription)

I encourage you to check out some of my other hyper-transcriptions, including:

Today’s hyper-transcription is from the short (but fairly technical) lecture that Alain Badiou gave at the European Graduate School in 2009 entitled “The Event as Creative Repetition.” I believe that my title is the better one: “The Three Fundamental Logics of Negation.”

There are three central principles within book IV of Aristotle’s Metaphysics. First, there is the identity principle which claims that a proposition is strictly equivalent to itself (i.e., is equivalent to p). Second, there is the principle of non-contradiction. The principle of non-contradiction claims that a proposition p, assuming that remains unchanged, can not be stated simultaneously with proposition non-p. Third, there is the principle of the excluded middle. The principle of the excluded middle states that either is true or else it is false. That is, either is true or else non-p is true. According to this principle, there can be no third possibility. The principles are arranged as follows:

The Three Central Principles of Metaphysics

Principle of Identity

Principle of Non-Contradiction

Principle of the Excluded Middle

Our first interesting observation has to do with the fact that the classical principle of the excluded middle assumes that the double-negation – the negation of negation – is absolutely equivalent to an affirmation. We’ve often heard this in our own day-to-day conversations. If you negate a negation then you supposedly end up with an affirmation. For example, there is a joke often told in anarchist circles which goes something like this: if you are anarchist parent (this implies that your political position is anti-statist) and you have a child then your child will be pro-state. The supposition here is one of double-negation because the child is negating the father’s anti-state position to arrive at an affirmation of the statist position. This view of things relies on the logic of the excluded middle insofar as the assertion of something, p, excludes the validity of the assertion of non-something, non-p. And so there is a choice that has to be made: either p or not-p. There is no other alternative available. So, when we write about classical logic we are always within this Aristotelian world-view which claims that a negation is never yes and no and always either yes or no. What is perhaps most interesting about this position is that the repetition of a negation exhausts that negation and reduces it back to an affirmation. Therefore, a negation of negation, or the repetition of negation, is a nullification of negation. For example, if my primary negation is to be anti-statist, by repeating the negation again I become statist; in this act, the original power of the negation is nullified since I am no longer anti-statist by any measure whatsoever:

Classical Form of the Negation of Negation

Primary Negation: Anti-State

Secondary Negation: Pro-State

We shall name this understanding of the logic of negation “classical”. However, classical negation is not the only logical possibility available to us. Most philosophers are already aware of this point. For example, Hegel very clearly argued that the negation of negation is not exactly equivalent to the immediate affirmation. Rather, the negation of negation is a principle of reflexive existence. We can therefore conclude that Hegel’s logical framework is not classical. In fact, we can derive several formal possibilities about the logic of negation. There are at least three logical forms of negation:

Logical Forms of Negation

Classical Negation

Intuitionistic Negation

Paraconsistent Negation

We ought to focus on the following two principles from Aristotle’s classical logic: first, the principle of non-contradiction, and; second, the excluded middle. Automatically, we can derive four possibilities:

1. (Classical Logic) Negation Obeys the Two Principles (Non-Contradiction & Excluded Middle)

2. (Intuitionistic Logic) Negation Obeys Non-Contradiction but Not Excluded Middle

3. (Paraconsistent Logic) Negation Obeys Excluded Middle but not Non-Contradiction

4. (Problematic Logic) Negation Does Not Obey Excluded Middle and Does Not Obey Non-Contradiction

The Fourth possibility – neither the excluded middle nor non-contradiction – implies that there is a destruction of any power of negativity whatsoever. For this reason, it can not, formally speaking, be a part of our discussion. It has no principle at all because it is a negation without remainder. However, the other three possibilities are consistent with propositional logic. The first principle is a formalization of the classical Aristotelian logic because it includes the traditional principles of the excluded middle and non-contradiction. The second possibility – the principle of non-contradiction but not the excluded middle – was constructed at the beginning of the last century. We find it in the work of the Dutch mathematician Arend Heyting. The third possibility – the principle of the excluded middle but not the principle of non-contradiction –  was constructed in the 60s by the Brazilian school. We name it para-consistent logic.

We can turn to the example of the violent war-time situation. In the framework of classical logic the conflict is structured in a binary or antagonistic way. For example, if a city is under military occupation then we can say that it is occupied by our own troops or else it is occupied by our enemy’s troops. It makes no sense, for classical logicians, to say that our city is occupied by our troops and the enemy’s troops. And, it makes no sense to claim that there is a third possibility, so that, logically, either our troops are occupying the city or else our enemy’s troops are occupying the city. In the framework of intuitionistic logic the conflict is not structured in such a rigid way. A city can not be occupied simultaneously by our troops and our enemy’s troops but the city will be called occupied if it is occupied by the troops of a neutral country. The result is that the claim that non-p, or, that no city is occupied by our troops, does not necessarily mean that every city is occupied by the enemy’s troops. So the excluded middle is not always valid and this makes war-time more complicated. Moreover, it makes war-time open to subtle negotiations and gradations in power. Finally, with paraconsistent logic, we can admit that a city can be occupied by our troops and also occupied by the enemy troops. Consequently, this implies that our troops are also not occupying the city to some degree. For example, in Stalingrad the soviet communique claimed that they occupied the city. This claim co-existed alongside the German report that they in fact occupied the city. There is no formal contradiction here because the one claim does not invalidate the other claim. 

However, what is most interesting about this paraconsistent logic is that we can maintain that the two reports co-existed even while maintaining the classical vision of the excluded middle: in a war, there can only be two active sides. The potency of the negation becomes increasingly weaker as we move from classical logic to intuitionistic logic and toward paraconsistent logic. This is because the destructive power of negation diminishes. In classical logic, the negation of excludes not only itself but also any other possibility concerning the content of proposition p. In intuitionistic logic, the negation of excludes itself but not the other possibilities that are located somewhere between and non-p. In paraconsistent logic, the negation of excludes that type of space between and non-p because we have the excluded middle but not itself. Thus, is not eliminated by the negation, is not destroyed by negation when the negation is paraconsistent.

For the moment we should return to Badiou’s distinction between things and objects. For Badiou, a thing is a pure multiplicity without any qualitative determinations. This implies that the laws of the world in which we find ourselves are not laws of things but rather laws of the qualitative determinations of things. They are laws regulating the determination or order between things in a given world. So, for Badiou, all laws are laws of appearing in the context of a singular world. So, a thing is not only a pure multiplicity, it also exists as an object in the world. This distinction is crucial – I have written a lot about it in my blog, so I will not spend a lot of time on it here. The distinction is between being-qua-being and existence, between pure multiplicity and the appearing of a multiplicity in the world, between a thing and an object. The point is that there is a different logic for objects and things.

The logic of being-qua-being, of things, is classical. It is classical because the composition of a thing as a pure multiplicity is extensional (see the wikipedia for the axiom of extensonality). Briefly, it means that the difference between two multiplicities can be defined on the basis of a single element of one of those multiplicities. So, two multiplicities are different if, and only if, there is some element of one multiplicity that is not an element of the other multiplicity. We can derive two classical principles from this: first: we can define as a multiplicity, and; second, we can define not-p as a set of all elements which do not belong to p. The result of this extensional claim is that not-p is absolutely different from p. The classical logic claims that there is nothing in common between and not-p. Here, the principle of non-contradiction is true because we can not find something which is simultaneously and not-p. Next, every element that is not in is in not-p, by definition. Not-p is a multiplicity of everything not in p. Clearly, there is no third possibility here because something is either in p or else it is not-p. The principle of the excluded middle is therefore true. Here we have the two classical Aristotelian principles: non-contradiction and the excluded middle

When are are at the level of appearing and existence, and not being-qua-being, then we have a different logic. A multiplicity can exist within the world with degrees of intensity. We have the principle of non-contradiction because a multiplicity can not not-be and be within the same world. However, the principle of the excluded middle is generally not true. For example, suppose that we have a maximal degree of existence within a world. A multiplicity appears in the world with a maximal degree of existence and another appears with a minimum degree of existence. Between these two extremes there can be all kinds of degrees of appearance in the world (though, it, of course, never disappears). If appears absolutely and maximally in the world and not-p does not appear at all, then it is not true that the only choice is between and not-p. The principle of the excluded middle is false in this situation. So, the logic of being-qua-being is classical and the logic of appearing, of existence, is intuitionistic.

being-qua-being – classical logic

being of existence – intuitionistic logic

being of event – paraconsistent logic

We have a third logical option here and it concerns the notion of an event. For Badiou, the logic of an event is neither reducible to pure being or to the laws of existence. This is why there are three logics and not only two; it is because there are three types of being. First, there is being-qua-being which is purely ontological and can be understood through the axioms of classical logic. Second, there is the appearance of existence of a being within a world, which can be understood through intuitionistic logic. Finally, we have the creative logic of the event within a paraconsistent logical context.

An event occurs when there is a sudden change in the degrees of existence for a great number of multiplicities that appear in a world. There is a movement from a minimal existence toward a maximal existence (incidentally, this is the definition that Badiou provides for a revolution, see here). For example, we could discuss the political existence of poor workers during a revolutionary event or the formal existence of abstract figures in a modern artistic event. When a multiplicity exists in the world with a very minimal degree of intensity then we can name that form of existence – “in-existence”. The poor worker in-exists. If we are asking the evental question then we are essentially asking: “what is the destiny of the in-existent multiplicities of the world after the event?” What becomes of the poor worker after the revolution? Moreover, what becomes of the lumpenproletariat, the truly in-existent? There are three possibilities:

  1. The force of change is maximal. In other words, the in-existent worker becomes maximally absorbed in the world. Among the consequences of the event there is a maximal intensity of existence for an object that was previously reduced to in-existence. The poor workers of the revolution, who were nothing before the revolution, become the heroes of the new world. In this case, the consequence of the event is classical. The consequence follows classical logic because the old world is formally reduced to the duality between the minimal intensity of existence (in-existence) and the maximal. This type of world has only two degrees of intensity and is therefore always classical. When the logic of the consequences of an event is classical then we have a true and genuine event. So, this point is worth highlighting; it is the consequence of the event which authorizes us to state that it is really an event.
  2. The change is intermediate. It is neither a maximal nor a minimal change. In this case, the in-existent population takes on an intermediate value somewhere between the minimal value and the maximal value. For example, the poor worker receives a reward for his struggle but is not considered, by any means, a new hero of a new world. In this case the logic of the consequence of the event or change is intuitionistic. If revolutionary politics implies that the logical consequence of an event be classical then reformist politics requires that the logical consequence be intuitionistic.
  3. Finally, in this case the force of the change can not even be perceived at the level of the in-existent. After the change, the degree of the existence of all of the in-existents of the world remains minimal. The poor worker remains a poor worker. The logical framework that we are dealing with here is paraconsistent because the consequences of the event are undecidable. Something happens, a rupture or a ripple in the order of a world, but everything remains exactly the same. There are no new values between the affirmation and the negation. The principle of the excluded middle is true but the principle of non-contradiction is false. This is a false event.

The argument that Badiou is making is somewhat counter-intuitive: a true change, organized through the consequences of an event, must be classical rather than paraconsistent – especially when the world in which that event occurs is organized according to intuitionistic logic. The point is that the consequences of an event can only affirm a classical logic within the context of an intuitionistic/paraconsistent opposition. It is because the world deploys an intuitionistic logic of negation that the triumph of classical negation, of a revolutionary event, depends upon the correct treatment of paraconsistent negations.

Given that a proposition behaves classically in a paraconsistent context, it validates the law of non-contradiction within that context. The principle of non-contradiction claims that it is impossible to have a contradiction. So, it is not true that we can have B as well as the negation of B. The proposition, B, is within a paraconsistent context. For example, this means that the revolutionary sentence is embedded within a reactionary context. The classical logic of the revolutionary sentence implies that one must decide between yes and no, either yes or no. But a paraconsisten context means that we have real contradictions. So, it becomes immediately obvious that we have to examine the possibility for a classical logic to be within a paraconsistent logical context. Formally, a given proposition behaves classical within an intuitionistic context if the proposition validates the principle of the excluded middle; it behaves classically within a paraconsistent context if the proposition validates non-contradiction.

Badiou provides another very interesting observation, or variation: whether classical, intuitionistic, or paraconsistent, all of the logics are written in the same basic lexicon. The difference between the logics really amounts to a difference of a logic of negation. There are more axioms that are shared between the logics than that depart from one another. This means that classical, intuitionistic, and paraconsistent logics share a fundamental logical background. It is only when we begin to develop an answer to the question of negation that we can introduce some differences between the logics. We find that there are three kinds of negation, and that the question of negation is, itself, a fundamentally philosophical question. It is not a question of experience, pure being, or even primarily of language. It is a philosophical question. Interestingly, for my own research at least, this means that Badiou calls philosophy what I call meta-ethics. In any case, every negative proposition, that is, every formula that begins with a negation applied to the entire remainder of the formula, behaves classically; it matters little if the context is intuitionistic or paraconsistent. The three logics propose different frameworks for negation, but, even granting this, if the proposition is absolutely negative at the end then it behaves classically. Every total negation is classical.

All of this brings us to Badiou’s understanding of the necessity of the classical logic of negation or revolution: when you face a decision – for example, the realization of a new political project or a new work of art – you are within the revolutionary imperative of classical logic. When you are approached with the necessity to reject a possibility that is presented to you, when, in order to continue to be faithful to the newness of your political construction, you have to resist another temptation, then you are within the revolutionary imperative of classical logic. The temptation is always to destroy your political project: to say “no” instead of “yes”. When this decision is forced upon you, then a “point” has been constructed.

A “point” occurs when the war-time situation is transformed into a pure decision. Either you do this or else you do that. If you do that, instead of this, then the whole revolutionary context is destroyed. Most often when a point dawns upon us we take the easy road, we make the easy decision, instead of struggling with the “yes” of the revolutionary impulse. The temptation is, for example, to be paraconsistent, within the intuitionistic logic, rather than classical. In other words, the temptation is to do that and also continue to do this. But the revolutionary situation demands that the impossibility of making that decision. You can no do both. The true temptation is therefore the temptation to have the best of both logics, the best of both worlds – to achieve compromises. It implies that we accept and also not-p; perhaps we accept a big part of but a small part of non-p – this is also a temptation. The struggle is to refuse temptation.

For this reason I conclude that all revolutionary situations are dogmatic insofar as that dogma refers to a classical logic of decision-making. The revolutionary is a doomed man – according to Nechayev – but it is the revolution that is truly doomed if the paraconsistent decision is made in place of the classical one.


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