The Politics of Logic: Badiou, Wittgenstein, and the Consequences of Formalism: Notes and Comments

My notes and comments from the first chapter of Paul Livingston’s fantastic book (so far) The Politics of Logic: Badiou, Wittgenstein, and the Consequences of Formalism.

Wittgenstein wrote in the Philosophical Investigations that “What has to be accepted, the given, is—so one could say—forms of life.” Theories of what he means oscillate between a conventional anthropologism of practice or culture and a naturalist reduction of language and meaning to scientific facts. Livingston wants to argue instead that the problem Wittgenstein invokes with the forms of life is not located in the question of the nature of lives or of their forms. He thinks it is what lies between the two terms: what it is for a form to be a form of life, what it means that something like form or forms shape a life at all. Rather than focusing on the “of” in forms of life, in what way does form inform life.
So, the aim of his book is to consider the relationship between forms or form and collective life under the conditions of an age that is determined by the technicization of information which is rooted in the formalization of language. This, he argues, requires an investigation in the consequences of formalism in two senses.

1. Considering the ways that life can be reflected in formal-symbolic structures and how these structures can illuminate forms of community and social/political association
2. Consider the effects of material and technological realization on these formal structures on the actual organization of politics.

Cantor’s definition of forms/formalism:

• “By a ‘manifold’ or ‘set’ I understand in general any many [Viele] which can be thought of as one [Eines], that is, every totality of definite elements which can be united to a whole through a law. By this I believe I have defined something related to the Platonic eidos or idea.”

If the Platonic idea captures the thought of the unity of the Many as One, then Cantor transfigures the problem in two ways.

1. Cantor/Frege grasped – the relationship of a set to its elements captures the relationship between a universal and the individuals that fall under it.
   1. Definition of a set is identical to the definition of a concept or general term. Likewise the relationship between an object and a property it has.
2. Cantor provides resources for addressing political philosophy – the question of the relationship of the One of the state, social whole, or community to the Many of its members

The claim here is not that a state, in order to exist, must in fact be a unitary structure. It is rather, more simply, that a social whole is always a many that is thought as a one, and the focus of the “political” interrogation of forms is thus on the question of the preconditions for this thought or for its actual effectiveness in praxis.

So through this book he will take the formal results of set theory and computation theory regarding the One and the Many to understand their implications for the nature and structure of political life.

He will argue that as set theory and formalism rise out of the Greek logos — language, word, meaning, rationality, reason.
The goal is to discern the basis of the phenomena of everyday human life in the broader phenomenon of logos.

“In the twentieth century, the material and historical “rationalization” of social life (for instance, in the widespread development and standardization of technologies and practices of communication, information exchange, and commodification) is in fact closely linked with developments arising from critical reflection on language and its formal structure or structures. Accordingly, both the concrete historical and the abstract critical consequences of formalism must be treated together if we wish to produce an analysis adequate to the most important social and political phenomena of our time.” Many existing analyses take into account the effects on social life of technology, progressive rationalization, and “instrumental reasoning.” But it is a substantial failing of many of these existing analyses that they do not consider, in any detail, the internal implications of the specific abstract and formal-logical structures that, on their own accounts, increasingly dominate social and political life. But if, as seems likely, the twentieth-century development of formal reflection on language and logic problematizes the very terms in which theorists have attempted to describe such an alternative modality of “lived” reason and reasoning, it may be that critical thought about technology and society must now continue explicitly in a formal mode, if it is to continue at all.

This produces a new question which (looking back) was one of the most significant questions of philosophical projects in the 20th century. This is the question of the formalization of formalism itself; the reflection of formal-symbolic structures within themselves, and the possibility of these structures coming to comprehend and articulate their own internal constitution and limits. This is sometimes called “metalogic” on the analytic side and “post-structuralism” on the continental side.

“One of my chief goals in the present work is thus to argue that these two strands of reflection on language—metalogical analysis on the “analytic” side, and post-structuralism in a deconstructive mode on the “continental”—can be allied, and thus can both be useful sources of critical reflection on the political implications of formalism.” He suggests that their combination provides a formally clarified understanding of the constitution and structure of political communities and their possibilities of changing and the internal dynamics of that change.
Aristotle suggests there is an essential link between the definition of being human and the possibility of politics. And, if the claim of logos on otherwise animal life is such, then a critical reflection on the “meaning” of this life must necessarily speak the language of logic as it does of politics.The critical reflection on formalism here is not applying some external logic to the political, not a logic of politics. It is instead a matter of making sense of the structure of logic in political dimensions, so a politics of logic.

The main question at stake isn’t how a community is structured out of the many who are united by common need or contract but rather understanding the logos itself as immediately the necessary form of any communication through language prior to distinct individuals agreeing on matters of fact or opinion.

Badiou suggested that all creative thought is a new mode of formalization and the invention of a form. . . so politics is also a thinking through forms. It isn’t the thought of arrangements or the thought of contracts or the good life, it is the thinking of form. So.. what form the good life takes. Or what form this arrangement will take. I think.

One of the projects of Livingston’s book is to examine Badiou’s system, which Livingston calls a dramatic interpretation of the implications of mathematical set theory for contemporary though about the structure of being and the possibilities of political change and transformation. It is a manifestation of the “politics of logic”. Badiou does run into some issues behind his claims, which Livingston will discuss, including having to reject continental structuralism and the analytic tradition.

Badiou has what he calls a “generic” orientation in thought. It is opposed to authoritarian and conservative orientation of traditional metaphysics and to the constructivist orientation that restricts being to the fixed law of an existing language.

Rather than rejecting one aspect of the One-All speak, Badiou is rejecting the possibility of describing the totality, the “all” of the set of all sets or the universe of all that exists. He challenges the completeness, but does not challenge the consistency.

The other orientation is the paradoxico-critical (PC) orientation. He uses figures such as Lacan, Agamben, Derrida, Deleuze, and Wittgenstein to show the features of this orientation and then juxtaposes them to Badiou’s generic orientation. All the PC thinkers stand in the legacy of the linguistic turn. All draw on high specified formal reasoning to document the critical and productive consequences of a consideration of the structure and boundaries of language. Instead of making the decision for the One of consistency and sacrificing the All of totality they choose the All and sacrifice the One.

He then moves again to Wittgenstein via Plato where Wittgenstein proposes that no rule can determine its own application. There isn’t a symbolic expression that by itself determines how itself is to be applied to new cases. The problem is that this leads to an infinite regress. So the understanding of rules as determined by their symbolic dimensions cannot succeed and must be supplemented by what Wittgenstein calls “mastery of a technique”: knowing how to apply a rule. This cannot be explicated in finite, symbolic terms but the knowledge can be applied to an infinite number of cases. So how is it possible to follow a rule? Wittgenstein’s answer is in the givenness of form, the form of life. It was of crucial importance to Wittgenstein that the use of language is always a realization of a capacity that can support indefinite repetitions of linguistic symbols and of justified uses. So a rule is not something that can only be followed once, but has an infinite number of adaptations and cases. It isn’t possible that there has been only one occasion that a rule was followed. So participation in practice or mastery of technique (knowing how to apply a rule) demands a social dimension of usage of a rule. But the problem of rule following is not just in communal standards, there is a deeper question of the very possibility of forming a community at all. It is a precondition to the application of a rule or law. To “agree” in a “form of life”, Livingston says, is not to agree in opinions or beliefs, or to be a part of a consensus based on communicative capacities of individuals, rather it is to “agree in language”.

The problem both Plato and Wittgenstein faced is the original positing of names, or the original force of law. It is the problem of the authority of whatever or whoever is entitled to pronounce or speak the law into force by some original declaration and pass judgement on a particular case. So the sovereign law of structure is the capacity of language to bind the instances of a life into the regularities of ordered sense. However, this position has been shown to be unstable as of late. A paradox arises: the act of instituting the legal order cannot be legal within that order itself. So the original institution and the continual force of a law depends on some founding gesture that was both illegal and exceptional with respect to the order that it founded. This exceptional position remains in the background of normal functioning societies even when obscured by a mystifying figure or feelings of unity of the political community. Contemporary regimes expand this one exceptional position to a more general “state of exception”, using Agamben’s language, which extends the claim of sovereign power indefinitely by citing the existence of an exceptional or emergency situation in order to reinstate the original indistinction between fact and law.

So.
The “critique of metaphysics” now has a political dimension. It operates to show the inherent contradictions at the center of any constituted political order and expose that order’s normative claims for critique. One of these is the sovereign’s claim to totality, the power to normalize and decide the legality of the various diverse events and facts that are treated as falling under its scope. But this isn’t critiqued from a transcendent position outside of the totality. The original basis of the power is recognized as an inconsistent position at the threshold of the order and what it excludes.

“Thus, the paradoxical topology of the sovereign position constitutes an original double bind between force and law. This topology can in fact be understood quite generally not only as the basis for specific empirically described political orders but for the normative forces of reason and measure themselves. It is determined, as we shall see, in a precise way by formally tractable results that arise from the profound twentieth-century logical and metalogical inquiry into totalities and their structure.”

Cantor and Diagonalization

Here Livingston argues that the metalogical insights and paradoxes it leads to of formal and syntactical structures of language are relevant to how we think about finitude, language, politics, and truth.

First, Cantor. Defined a set as “many which can be thought of as one.” His theory of sets presupposed that unity of a concept can be theorized as a totality. This means that anything grouped together by language or concept is a unified one. So an infinite series of natural numbers is a single completed set.

He provides the bases for the mathematical concept of the infinite to be separated from religious/mystical concept of the infinite. He treats the religious concept as transcendental and unthinkable. The way to conceptualize infinite is through diagonalization. You think of an infinite number of elements in a system, the totality of which exhausts the system of comprises all elements with a certain property in it. (e.g. The infinite set of natural numbers or the (infinite) totality of sentences in a language.)

Diagonalization takes this totality and works on it to produce a new thing that is:

1. Formally a member of the totality (has the right properties to be a member)
2. Demonstrably not the same as any single elements in the system.

This shows that the power set is bigger than the original set. It creates a hierarchy of transfinite sets. Each infinity is larger than the last. This suggests a radical change in theological categories of total and absolute. Paradoxes developed that seemed to show it is impossible to conceive of such a totality. One such paradox is Russell’s Paradox.

Russel’s Paradox, Godel’s Incompleteness Theorem, and Parameterization

Russel’s paradox is concerned with the possibility of constructing sets or groups of individual objects. The operation of grouping or collecting individuals and universal concepts or general names is the fundamental operation of linguistic reference. So this paradox has implications for language and representation. Russell’s paradox was set specifically against Frege’s “Universal Comprehension Principle”:

• For any property nameable in language, there is a set consisting of all and only the things that have that property.

Russell’s paradox, the liar paradox of Epimenides, and the paradox of the Creten all contain self-reference: All these paradoxes attempt to say something about totality and then to create a set that is both part of the totality and not part. So by way of self-reference, language invokes a radical paradox of non-closure. In both Russell’s paradox and Godel’s incompleteness theorem a problematic element (Russel’s set/Godel’s sentence) is produced by combining totality and reflexivity. This element reflexively captures the total structure of the whole system of which it is a part of at a specific point within the system.

Formalism sought to reduce mathematical reasoning and inference to a purely mathematical process. Russell noted that the prohibition of formation of totalities that include members defined in terms of themselves will lead to problems with the formulation of principles and descriptions that otherwise are or seem natural. So, this leads to Russell’s theory of types. A set can be a member of another set but only if the containing set is a higher type or level. This means that a term can make reference to other linguistic terms but cannot reference itself. *This type of solution underlies all subsequent historical developments of set theory after Russell.* The intuition behind Russell’s types and standard axioms is “the standard iterative conception of sets” meaning the universe of sets is inherently hierarchical. Almost immediately Godel’s incompleteness results in a demonstrated need to supplement internal, syntactic description of language with a dimension of external meaning or reference outside the system under consideration.

So the requirement of irreducible and unlimited hierarchical levels to let sets only to have members lower than themselves. Graham Priest calls this Parameterization. This solution relativizes claims about language, truth, and reference to indexed types or levels. The price of this is the inability to talk about the totality of all sets or formulate in any terms the logic of language that is complete in the sense that it can refer to anything that exists.

• The Pros: Parameterization preserves consistency in the face of paradox and contradiction.
• The Cons: Constructivist picture of formation of infinite sets demands reference to agent capable of completing infinitely extending processes.

So we are still left with a question of the possibility of unity of many as one and the formulation of a unity or totality at all in language, truth, history, power, etc. These paradoxical situations are treated by Graham Priest as limit-paradoxes which necessarily arise at the boundaries of thought. it is possible to generate one whenever 2 formalizable operations are possible:

1. Closure – formalizes conditions necessary for an element to be a member of a given totality. (Draws a boundary)
2. Transcendence – any operation that given a totality can generate an element outside of this totality.

If it satisfies both 1 and 2 then it is possible to generate a contradiction. Priest says this kind of contradiction is at the limits of thought and language is a formal version of the paradoxes philosophers are concerned with. So it is possible to use parameterization to avoid paradox. We can only talk about truth in English given a metalanguage or we can only name an event after the event has taken place. But this is inadequate. We improve a situation by reconsidering what is involved in the underlying paradoxes. Paradoxes don’t demonstrate the incompleteness of language. They give us a choice between incompleteness and inconsistency. On the level of formal languages we can save consistency of systems by ascending up a hierarchy of metalanguages or Priest says modal inconsistency within a self-contained formal language by means of dialethic logic. One that tolerates contradictions in certain cases.

Symbolic language must consist of finite corpus of symbols, but be capable of infinite iterations and applications, just as Wittgenstein suggested. There is essentially no alternative. The structure must be finite or we have no sense of what we are talking about when we mention its rules, etc.
*Things are the same with respect to critical considerations of the structure and limit of communities. Here as consideration of problem of force, authority of rules, laws, and norms demanded are an essentially formal reflection om the relationship of finitely stateable rules and an infinite iterability of its consequences.

If parameterization and the decision for incompleteness if not an option in language then we are forced to choose for inconsistencies at the limits of any thought or language has the capacity for self-reference. This forces the position that there are inherent contradictions/inconsistencies involved in the practice of speaking language itself. These contradictions are a problem for enclosing totalities with the hope of a complete and consistent system. These contradictions also give opportunities for the critical thought that challenges any project. Since Kant, critique has been understood as the practice of tracing the closure of totalities. This comes from Kant’s project of limiting knowledge to the boundaries of the experienceable. This lays the groundwork for Wittgenstein to be able to make the claim that in order to draw a limit to thinking we have to be able to think on both sides of this limit, thinking that what cannot be thought.

Both Derrida and Wittgenstein were reflecting  on the practice of language and the possibility and force of rules because these define the relationship between the (finite) sign and its (infinite) possibilities and contexts of use. This Livingston calls the paradoxico-critical position.

Agamben’s Coming Community

The thinkers mentioned earlier elucidate the fundamental issues of totality and reflexivity in thought and language. What do these have to do with social structures or communities? Here, Agamben provides an answer. His 1990 book The Coming Community sees the underlying basis of a community is the possibility of grasping and using the paradoxes of linguistic meaning. He introduces the term linguistic being as the capacity of any object or phenomenon to be named, its entry into the totality of language itself. Shifters (I, it, the, etc.), words that name, show the problematic self-reflexivity in language. Shifters are a representative that is fully individual but also stands for the whole universal class. (e.g. When I say “I” the referent is different than when you say “I”.) These are a kind of  “exclusive inclusion”. It is a demonstration of the general structure of inclusion within normal parameters, but operates by excluding the exemplary in the moment of demonstration from a normal case.

• cf. Russel’s theory of types and hierarchy of sets
• cf. also Priest’s limit paradoxes.

In crossing both the universal and particular they show the power of naming things–of grouping them together. Agamben sees the power of grouping whose basic ambiguities are shown by exception as the underlying basis of naming and of the force of law itself. The ability of language to subsume individuals under general concepts is also the foundation for the application of laws or rules. In both language and law, moving from the general to the particular depends on the power of grouping via the system of rules (of language or otherwise).

In his books Homo Sacer and State of Exception, Agamben makes the connection between sovereignty and exceptionality. The sovereign power defines political space by its power of deciding the exceptional cases. Normal applications of laws depend on some semblance of what is the normal order, so the sovereign must decide what counts as normal or exceptional. This also implies that the Sovereign can decide when to suspend the normal order. They decide when an exceptional case of facts or “emergency” circumstances justifies the suspension of the normal order of law based on the sovereign’s original positions. This places it both inside and outside the order of law that it founded.

A quintessential example Agamben provides is of Hitler’s suspension of the articles of the Weimar Constitution in 1933 which protected the personal liberties of subjects. This essentially created the Nazi state. Agamben suggests that once it occurs the suspension of ordinary rules of law is usually irreversible. A total or partial suspension of the rule of law for a state of exceptionality is no longer limited to states that are explicitly totalitarian but is one of the essential practices of a wide variety of states–even ones that describe themselves as democratic.

This paradoxical structure of the sovereign, determining the distinction between normal and exceptional or law and fact, is formally identical to the Russell paradox, Livingston says. The sovereign, in choosing what is normal or exceptional, both includes and does not include itself. It rests on a paradox.

Badiou and the Event

According to Livingston, Badiou’s application of the formal methods to the theorizing of the “Event”, the transformative eruption of the essentially unforeseeable new into a given, determined situation, is one of the most important outcomes of his thinking. In order to make sense of novelty, Badiou wants to model the ontological regime of being so as to develop a schematism of what takes place outside of this regime. This schematism will then be the schematism of the event itself. In Being and Event the fundamental axiom of Badiou’s project is the identification of ontology with mathematical set theory. The axioms of set theory are a formal theory of whatever is. This is essential to his description of the form and limits of being and to defining the possibility of the event which, while different than being, can intervene within being to bring about change. In Logic of Worlds, he presents a formal take on the linguistic establishment and transformation of the boundaries and structure of particular situations which he calls “worlds.” Here, transformation of the structure of a particular occurs. This time an ontologically errant set-theoretical structure allows for retroactive change through revealing what was previously invisible. Badiou sees formalism as rendering the infinite mathematically (and hence ontologically) thinkable. Cantor’s work allows for the size of various infinite sets to be compared. This impacts our ability to think of the one and the many.

Badiou describes set grouping or unification as a result of “counting as one”. This forms an indifferent multiplicity into a structured one that can be counted or presented. Prior to this counting, Badiou calls these inconsistent multiplicities. They cannot be thought of or conceived mathematically or ontologically. Badiou says that fundamentally there are only multiples and multiplicities (sets). They can be counted as one or grouped into ones. But the One-All, the universe that results from grouping everything that is together, cannot exist. Thus he opts for the choice that the universe, described by language, is incomplete.

He then makes a distinction between the presented and the represented. He utilizes sets to make this distinction. The set containing Alain, Bertrand, and Cantor has 3 elements in it but has 8 subsets (Alain, Bertrand), (Bertrand), etc. and empty sets. All 8 subsets combined into a unity is a power set of the original set of (Alain, B, C). It is always larger than the original set. The power set re-counts or re-gathers the elements of the set. This recounting of power sets is representation. The distinction between the initial set and the power set is the same as between presentation and representation. Further, he describes the initial set as “the situation” and the power set as the “state” of the situation (contains whatever, presented in the initial situation, can again be regrouped and re-presented as a one in representations). The number 5 for example, contains the elements of 0, 1, 2, 3, 4. The number 4 contains 0, 1, 2, 3. Each of the numbers is present in the initial situation and the state of the situation is just the regrouping of the elements (0, 1), (1), etc. Nothing new emerges between the presentation and the representation. This is a “natural” situation where the conditions for change aren’t there as representation can only reorder elements that were there initially.

This is not the case in “historical” situations. In these situations the elements can have other elements that are not already a part of the situation, this is a “singular” term. While it is presented in the initial situation, it will not be represented in the situation’s state. Through the regrouping, elements that weren’t present in the initial situation can come be revealed. Badiou calls these “excrescences”.

Livingston notes that the “state” metaphor is deliberate on Badiou’s part in making a political statement. He suggests that representative politics, even in their most democratic forms, works fundamentally not by representing each individual that is a part of the state, but one removed from them. It only represents them as subgroupings or constituencies. This makes the state excessive to the individuals whose interests it is supposed to address. Its excess is of representation over presentation. It is the names and concepts that define group and constituencies over the members of the group themselves. So. Any sort of real intervention in politics is an intervention on names as much as it is on its bearers. Genuine political change does not result simply from the “collective” action of individuals, but rather from the formation of fundamentally new groupings and political identities. Any possibility of genuine change is dependant on hidden possibilities of the names and concepts that are already circulating to undermine the logic of the existing situation. This new development that is an intervention on what will have happened determines what has happened. This is called an “event”. This paradoxical nominalization breaks with ontology and summons genuine novelty “from the void.”
The empty set is the simplest element that assures there is something in existence. So, Badiou says that the empty set ties set theory to the basic assumption of being. This is the name of the void. The name depends on a fundamental act of self-reference. This void reminds us that self-reference or self-calling out is the first element the is the foundation for everything else. The ability to assert “there is. . . “, to self-reference, is the most essential characteristic of the event.

To illustrate this Badiou uses the example of the French Revolution. “The French Revolution” refers to a variety of individual events, statements, things, etc. that occurred between 1789-1794 in France. The ability to determine the various occurrences and circumstances as counting as one in the unity of an event depends on the exact moment where the revolution names itself. IT calls itself into existence as the event it will have been. The event’s occurrence depends on its grouping together  or recounting as one the various elements of the situation in which it intervenes (the event’s “site”). The event then, is the set composed of all the elements of its site and itself. The ability to include itself allows the event to bring novelty to the situation that wasn’t seen before.

Another way I think you can think about this is via “love”. Various individual situations occur between two people, a held gaze, flirtation, kissing, etc. “Love” might exist but it is not discernible until one person thinks about it. That moment, the moment they realize, “I’m in love”, the Event has been named. Novelty is introduced and all of a sudden all of the discrete situations that occurred now are subsumed under the event called love. Badiou says that “only an interpretative intervention can declare that an event is presented in a situation.” This summons the event into existence from what seems to be the void but is “the arrival in being of non-being, the arrival amidst the visible of the invisible.”  Now introduced, the event causes various consequences. Tracing what these consequences Badiou terms “fidelity”. The operator of fidelity, who discerns the consequences, is called the subject. Once an individual first realizes they are in love and it exists, then they can trace the consequences of it (e.g. gazing at them, kissing, etc.).

Badiou says that within the constructivist vision of being there is no place for an event to take place. The universe of what exists is limited to what can already be named in what already exists. The constructivist vision allows for the construction of a universe of sets which are infinite but are restricted to the condition that each existing set must be constructible out of already existed ones by means of the predicates already defined in a language. It prevents the construction of self-membered sets.

On the surface Badiou’s attempt to articulate novelty beyond what can be said puts him in a precarious position just as Wittgenstein did by speaking of what he says is unspeakable or by tracing the boundaries of what is sayable to indicate what lies beyond the sayable. So, Badiou separates the formalism of math from language itself thereby attempting to formalize what is sayable of being and what lies beyond by means of the abstract (the non-linguistic). Where language cannot speak, the abstract transmissions of the formalisms of math can display both the structure of the sayable and the events which lie beyond the sayable. Livingston recognizes though that Badiou doesn’t acknowledge that Russell’s paradox or Godel’s incompleteness theorum allow us to choose between incompleteness and inconsistency. Russell’s paradox doesn’t force us to reject the “One-All” that Badiou claims it does. Livingston suggests that the effect of the paradox is to split the One-All into two different hypotheses and forces us to choose between them. Either we can reject the “All” of totality and preserve the “One” of consistency (which Badiou chooses) or we can choose the All of totality and reject the One of consistency. The second option is a different orientation that Livingston will explain.

Four Orientations of Thought

Badiou offers three possible orientations in thought in his book Briefings on Existence and Livingston adds a fourth to that list. The orientations consist of

• Paradoxico-critical (PC) – Any position that, recognizing reflexivity and its paradoxes, nevertheless draws out the consequences of the being of the totality, and sees the effects of these paradoxes always as operative within the One of this totality.
• Generic (G) – Any position that, recognizing reflexivity and its paradoxes, denies the being of the totality and sees these paradoxes as traversing an irreducible Many.
• Criteriological/Constructivist (CC) – Any position that attempts to delimit the totality consistently from a stable point outside of it.
• Onto-Theological (OT) – Any position that sees the totality as complete and consistent in itself, though beyond the grasp of finite cognition, which is located simply within the totality.

PC is distinct from CC which delimits being through an investigation of the fixed structure of language. PC takes account of the paradoxes of self-inclusion to trace the fundamentally paradoxical structures of limits up to the paradoxes involved in the fact that language appears in the world at all. PC affirms (even if it leads to paradoxes) that natural languages like English contains all the resources for talking about its own structures. CC attempts to describe the boundaries of a language by describing the rules invoking a hierarchy of metalanguages. PC and G differ on the status of totality. G saves consistency by denying totality and PC affirms an inconsistent totality by showing the inconsistencies that arise when language ventures to speak the whole as One. The point of PC is to show how the most rigorous One essentially becomes many once it passes through the unifying function of language which produces the gulf between symbol and reference.

Having looked briefly at the the reflexive possibilities and the limits of formalism, Livingston sees a few resulting consequences that are further explored in his book. 

First, a new kind of thought about political structures as evident in Agamben and Badiou (and retrospectively in Wittgenstein and Derrida). The aim of this thought is to demonstrate the implications of the phenomena of inclusion, representation, organization, and desire for consistency and totality. This new political thinking of logic offers to comprehend and interrogate much closer the complex structures of power that operate in post-industrial cultures. He suggests that the primary manifestations of strategies and claims of power in the age of globalization are no longer single “totalitarian” or onto-theological figures of the One, but rather the diffuse, plural flows of information and capital linked within networks of corporate, technical, and military power. These networks have depended on innovation in computational and communicative technology that are themselves the outcome of the logic that developed in the 20th century.

Second, are the considerations of the totality which is reflected and placed at a specific point within it through paradoxes of self-inclusion and self-reference. Acknowledging this paradox is recognizing the groundlessness of the naturalist and conventionalist account of the origin of language and norms. It reveals a radical alternative to the debate between the (leftist or constructivist) politics of contingent historical conventions on one hand, and the (rightist or onto-theological) politics of an assumed “human nature” or divine dispensation on the other. Thus, if the politics of logic and trace the foundation of contemporary thought on the suppression of the paradoxical position of reflexivity then it can be overcome through traversing the paradoxes.

The notes for chapter two can be found here.