**I. Introduction**

Reading Hegel with Lacan, as Žižek is wont to do, raises the question of Hegel's relation to formal ontology. Formal ontology is a post-metaphysical discipline concerned with the formal conditions of subjectivation, the exception, the (formal) universal, decision and its relation to emergent logics, and so on. At first pass, and in more traditional readings, Hegel appears as a philosopher who repudiates form, who moves from form to content in an immanent process of successive internalization. On the other hand, Hegel's idealism places him rather close to formal ontology, for what else could formal ontology be but an ideal moment emerging immanently from material processes? Indeed, Alain Badiou, one of formal ontology's most ardent and sophisticated defenders, could to an unsympathetic reader be considered an idealist himself. In fact there does seem to be something of the ideal, something of metaphysics, in formal ontology. This ideal moment within materialist dialectics necessitates a comparison to Hegel as master of the “idealist” dialectic.1

The argument will proceed in three sections. First, the Hegelian unity of form and content can be reasonably interpreted as intelligibility rather than idealism. Intelligibility satisfies the Parmenidean thesis of the identity of thinking and being, which can reasonably be construed as a necessary axiom of formal ontology. In Hegel's dialectic of form and content, he reaches the absolute form which flips over into intelligibility. Second, Badiou reads Hegel's philosophy as non- or anti-formal, as claiming a non-formal generative immanent development bolstered by the “Axiom of the Whole” while nonetheless relying on a moment of decision, an exteriority that is supposedly intolerable to dialectical logic. Third, this critique is called into question when we read Hegel as a formal ontologist, showing that dialectical logic does not rest solely on interiority or even interiorization, but requires a moment of decisional non-relation. The difference is that for Hegel speculative non-relation is formally implied but not mediated (and furthermore not necessitated in its actualization) by the contradiction of the categories, by their relation.

**II. Formal Ontology: Intelligibility & Logic**

Formal ontology, as formal, necessitates a focus on form. But this is not the form thought by Verstand. In Hegel, we proceed from form as simple exclusion to absolute form which is finally determined as intelligibility. The centrality of intelligibility in turn necessitates a comparison between mathematics and logic, an important philosophical debate revived by Badiou. Hegel, however, subordinates both formal logic and mathematics to a kind of meta-logic (the dialectic). The meta-logic Hegel has in mind requires the positing of the Whole, which will take us into the next section.

Generally speaking, form in its first determination is the simple exclusion of content. Form and content are, on the level of Verstand, mutually exclusive concepts. Insofar as we are looking at something's form, we are explicitly not looking at its content. This is form as indifferent to content, as an abstract container of content. This is the sense in which we say that form is “empty”, as in formal logic when it can be “filled” by any content whatsoever. Symbols like “P” and “Q” stand in for any propositions whatsoever, with the caveat that they are “simple” or “atomic” propositions. Already here in this caveat we find the impossibility of sticking strictly to form as exclusion of content.

This is the essence of the dialectic of form and content, or really any dialectical development in the Science of Logic, that the opposition develops into a contradiction and subsequently the terms become moments of some further concept.2 The dialectic of form and content progresses to the Absolute, as the (speculative) identity of form and content: “The distinction of content and form itself has also vanished...” (Hegel, Science of Logic 551). This vanishing can be understood as an answer to the question “what does the Absolute manifest?”. The answer, now that we have the unity of form and content, is that the Absolute just manifests itself. The question as to further determinations remains misguided, if one is looking to some external criterion. Of course, the Absolute will be further determined in its immanent development, into the Notion, but from there onward form and content remain indistinguishable.

Once we get to the Notion, there is no question of adequation to the object: once we truly have a Notion, in some sense, we also have its object, we have its reality:

The derivation of the real from it if we want to call it derivation, consists in the first place essentially in this, that the Notion in its formal abstraction reveals itself as incomplete and through its own immanent dialectic passes over into reality; but it does not fall back again onto a ready-made reality... (Hegel 591)

The Notion, properly conceived, passes into reality. Here we recognize the earlier dialectic of form and content on a higher level, with that of Notion and reality. As if to emphasize that we have not left form behind, but have only reached its truth, Hegel writes: “For this reason, this form is of quite another nature than logical form is ordinarily taken to be. It is already on its own account truth, since this content is adequate to its form, or the reality to its Notion” (592). The dialectic has not proceeded in a vulgar manner from form to content in successive internalization, but has finally at the end of essence resulted in a kind of absolute form, which does not exclude content but which does not simply “contain” it either. Instead of form as indifferent to content, we have the form-content distinction as itself indifferent, but trumped by a form more formal than form itself. This is precisely “intelligibility”.

We will call the “principle of intelligibility” the necessity of intelligibility, which implies the imperative that thought cannot rest on a contradiction. This is not to say thought denies contradiction, as in classical logic. But contradiction is a productive occurrence: A & ~A implies something new. Importantly, intelligibility itself is first of all on the level of method (or metalogic), meaning it can be made unintelligible if it fails. If we are unable to move on from a contradiction, the principle of intelligibility itself has been made unintelligible, since it comes into contradiction with the reality of the situation. Since “the essence of intelligibility is synthesis”, the so-called synthesis is what once again restores the principle of intelligibility (Birchall 286).

And this intelligibility necessarily depends on the vanishing of the form-content distinction. The dialectic can thereby be understood as an immanent logic of intelligibility: “A logic of intelligibility must be intelligible to itself in its own terms. But this is possible only if it is free from the separation of form (validity) and matter (truth)” (Birchall 286). Since here we have a reference to formal logic (validity and truth), we should turn to the status of logic within formal ontology, as well as its difference from mathematics.

Lacan himself, in Seminar XX, more or less inaugurates the project of formal ontology with his use of formal logic to express the formulas of sexuation.3 But he also came to use topology in the project. To him, this was perhaps not an exclusive choice, but with Badiou it takes on the character of a fundamental decision. While Badiou affirms mathematics, in particular a certain axiomatized set theory, as the paradigm of intelligibility, on the basis of which he declares the identity of thinking and being, Hegel clearly affirms logic: “...logic is of course a formal science; but it is the science of the absolute form which is within itself a totality and contains the pure Idea of truth itself. This absolute form has in its own self its content or reality” (592). Hegel is here designating logic as the science of intelligibility. But we must be careful what sort of logic we mean by this. Formal logic as usually understood takes a place in the derivations of the Science of Logic, namely in the Notion. But there is another kind of logic, namely dialectical logic, by which the categories progress. It is this logic which is the science of intelligibility, not Aristotelian syllogisms (the formal logic of Hegel's day). What are the implications of Hegel's affirmation of this logic, as against mathematics?4

We can illustrate these implications with a reference to the difference between Hegel and Badiou. Hegel provides a logic of intelligibility which actively produces intelligibility from the unintelligible. Since this logic is furthermore immanent, what Hegel provides us with is the essential operation of formal ontology, formalization. Badiou, on the other hand, already begins with mathematics. In that sense, he does not give us a universal method of progressing from the non-formal to the formal. Or at least, that progression is for Badiou extrinsic or contingent, whereas for Hegel it is necessary or intrinsic. But in the Science of Logic, the nature of this “necessity” is logical or formal. One way to understand this is to consider the difference between mathematics and the logical structure necessitated by mathematical proofs. To formalize new elements is to construct a (logical) proof (of mathematical statements). Hegel proposes that to find intelligible implication or formal entailment we must look to logic, and in the context of philosophy this ought to be dialectical logic.

**III. Decision & The Axiom of the Whole: Badiou's Reading of Hegel**

Badiou has chapters critiquing Hegel in both Being and Event and Logics of Worlds. In this section we will focus on two objections in particular: Hegel's supposedly foundational “Axiom of the Whole”, as found in Logics of Worlds; and the deduction of the good qualitative infinity, as found in Being and Event. We will summarize these two critiques and then analyze them to lay the groundwork for the following section, which will give us the tools to respond to Badiou.

In Logics of Worlds, Badiou's argument against Hegel focuses on the Whole. It is for Hegel inconceivable that essence does not appear – it might not appear here or now, but it must appear somewhere in the Whole. This is a reiteration of a rather pedestrian argument against Hegel, only here it is given a more sophisticated tinge by reference to Russell's paradox (the Whole cannot exist for Badiou because it would be contradictory in set-theoretical terms). In any case, it is necessary for Badiou's critique of Hegel, since he aims to show that Hegel decides for the Whole (by axiomatic decision) and then subsequently breaks that axiom by introducing something external to the Whole. Because Hegel's supposed dependence on the Whole or totality is well-known, we will focus primarily on Badiou's other critique. We will, however, respond to both these critiques in the following section.

In Being and Event, Badiou summarizes Hegel's project: “The ontological impasse proper to Hegel is fundamentally centered in his holding that there is a being of the One; or, more precisely, that presentation generates structure, that the pure multiple contains in itself the count-as-one” (161). Hegel begins, in other words, with the decision that there is a whole (a One), which somehow legislates the count-as-one. The count-as-one is, in Badiou, the extrinsic function which converts (or captures) inconsistent multiplicity as consistent multiplicity, the proper expression of which is set theory. We move from formless “stuff”, the inconsistent, to individuated “ones”, which however remain multiplicities. For Badiou the count-as-one is not inherent in pure multiplicity. Pure multiplicities might not go into their count-as-one without remainder. For Hegel, on the other hand, there is a necessary relation between multiplicities (or objects, or things) and their articulation by thought. The Notion, as absolute, does not however stoop down to become the thing. Rather the thing reaches up to become adequate to its Notion. This motion is just that from unintelligible, analogue of inconsistency and contradiction, to intelligible, analogue of consistency and reign of the count-as-one.

We have thus determined the alleged difference as one of the extrinsic and the intrinsic. The One is just interiorization, the ever-larger passing-beyond where even passing-beyond has been passed beyond, brought to its conclusion: “It would not be an exaggeration to say that all of Hegel can be found in the following: the 'still-more' is immanent to the 'already'; everything that is, is already 'still-more'” (Badiou, Being and Event 162). This is Hegel's critique of fixity or reification. To hold that the “already” has only an extrinsic relation to the “still-more” is to remain on the level of Verstand. It is to deny the dialectic, since it denies movement and therefore contradiction. Elaborating on the intrinsic in Hegel, Badiou writes:

In a subtractive ontology it is tolerable, and even required, that there be some exteriority, some extrinsic-ness, since the count-as-one is not inferred from inconsistent presentation. In the Hegelian doctrine, which is a generative ontology, everything is intrinsic, since being-other is the one-of-being, and everything possesses an identificatory mark in the shape of the interiority of non-being. The result is that, for subtractive ontology, infinity is a decision (of ontology), whilst for Hegel it is a law. (Being and Event 163)

This is an extraordinarily rich passage. It accuses Hegel of vindicating intrinsic-ness through the interiorization of non-being or the negative. The One is already the other, the next one. Badiou alleges that immanent intrinsic development is inconsistent with decision, that the two cannot go together: in Hegel, “everything is intrinsic”. It is Badiou's claim that generative ontology makes Hegel unable to really theorize quantity, and therefore also to understand (the good) quantitative infinity.5 This will then mask the necessary decision or moment of extrinsic-ness that marks the step from the good quantitative infinity to qualitative infinity. According to Badiou, Hegel both requires and cannot make use of the extrinsic. It is a notable difference: for Hegel, “everything” must be intrinsic, but Badiou claims he needs only “some” exteriority.

Badiou's argument here can be summarized in the following manner: In the transition from quantitative to qualitative infinity, Hegel makes a return to pure presentation. The “something” passes from bad repetition to the pure presence of itself to itself as repeatable. It “voids” its other and is only self-related. The quantative One is thus the anonymous One, indifferent to difference: “It is not that it is indiscernible: it is discernible amidst everything, by being the indiscernible of the One” (Badiou, Being and Event 167). This then would be the difference between truth as necessary adequation in the One and Badiou's truth as generic indiscernibility, that the former is a form of unrestricted universal that not only is addressed to all but is necessarily in all.

This anonymous One is now the primary determination of quantity. But there is a fundamental disjunction between the two orders according to Badiou: “Quality is infinite according to a dialectic of identification, in which the one proceeds from the other. Quantity is infinite according to a dialectic of proliferation, in which the same proceeds from the One” (168). These two are not the same thing, but Hegel connects them simply by a return to pure presentation and then a nomination of both of them as “infinity”. To Badiou, this illicit nomination is the only term which bridges the abyss between the two orders. The nomination thus secretly constitutes a decision on Hegel's part, one which leaves quantity (and with it mathematics and infinity itself) behind forever: “On the basis of the very same premises as Hegel, one must recognize that the repetition of the One in number cannot arise from the interiority of the negative. What Hegel cannot think is the difference between the same and the same, that is, the pure position of two letters” (Badiou, Being and Event 169). Again, for Hegel the same and the same cannot be different because of the determination of quantity as the anonymous One. Precisely as indifferent to difference—but in all the wrong ways, according to Badiou—quantity cannot think its own difference, hence it returns to simple presentation and to quality. But the position of the letter, the difference in position of two identical letters, is not quality, or at least quality does not capture what is quantitative or mathematical about it. What Badiou is saying, in plain terms, is that Hegel cannot think a truly mathematical notion of numerical identity. Therefore, there is in Hegel no good mathematical infinity.

The implications of this are vast. It is first of all a decision, extrinsic, in what is supposed to be a process of interiorization. It is imputed from the “outside”, as it were. The dialectic should have been forced to stop there, foundering on the rocks of mathematics. On the other hand, that decision also explains Hegel's preference for (meta)logic, since it is precisely a domination of quality over quantity. Though we might think formal logic to be abstract and indifferent to quality, dialectical logic (which is itself a strange sort of immanent meta-logic) has decided decisively for quality. This is why, perhaps, the dialectic cannot be straightforwardly (or easily) formalized into a cousin of “formal logic”, and indeed, dialectics always has something of the qualitative in it, no matter what category it unfolds. It is doubtless harder to see the quantitative as a continuing dialectic in the Science of Logic. The decision implies, in other words, that for Hegel mathematics cannot be the language of being, as it is in Badiou. Mathematics, insofar as it affirms its pure presence, becomes rather for Hegel something qualitative. For Badiou, in contrast, mathematics is just pure presentation.

To explicate this claim, we now turn to what is undoubtedly the greatest difference between the two thinkers, what determines the very nature of their different relations to formal ontology, namely whether formalization is a function of contradiction or subtraction. This exploration will determine the precise implications of Badiou's critique: Is it possible to have a formal ontology based not on exception, but on contradiction? And can Hegel be the basis of such an ontology?

**IV. Form Against Form: Contradiction & Exception**

Intelligibility is an important concept in Hegel. His vision of intelligibility is intimately related to the dialectic, to contradiction which is sublated by the movement “towards the Whole” (or at the very least towards ever more relations).6 But the logic of contradiction and the logic of exception, that logic proposed by Lacan, (partially) Žižek, and Badiou, do not immediately coincide. Typically, these are considered to separate and mutually exclusive logics, at least in the context of formal ontology (obviously different applications might call for different frameworks, but that is not at issue). Formal ontology was conceived by Lacan as based on the exception, for example in the formulas of sexuation the universal is founded on an exception. Žižek offers us a reading of Hegel that (implicitly) fits the two logics together in sometimes-unintelligible ways, whereas Badiou has divested contradiction of its foundational role in philosophy. It is my claim that Hegel does not in fact oppose contradiction and exception, but utilizes both within his dialectical logic. Both, it must be remembered, are intelligible categories in this context. We will now attempt to explicate this claim.

We have already seen how contradiction relates to intelligibility in Hegel. Žižek summarizes as follows:

'Contradiction' is not opposed to identity, but is its very core. 'Contradiction' is not only the Real-impossible on account of which no entity can be fully self-identical; 'contradiction' is pure self-identity as such, the tautological coincidence of form and content, of genus and species, in the assertion of identity. There is time, there is development, precisely because opposites cannot directly coincide. (629)

What is being referred to here as “contradiction” is thus the very principle of intelligibility. This suggests that we can interpret contradiction as itself a formal process emerging from the immanence of material reality. Interpreted in this way, contradiction is not “in things,” in material reality simply as such, but emerges from it as a formal process. Transposing to the language of essence and appearance, Žižek writes, “The distinction between appearance and essence has to be inscribed into appearance itself” (37). This is the emergence of form or truths immanently, a non-metaphysical theory of form..Whereas on traditional readings, this sort of thing would be interpreted as idealism, contemporary readings could conceivably interpret this as materialism, in particular formal ontology. This sounds a lot like the exception, but is it?

The exception is, in Badiou, that which is in reality but not of it. It does not refute the materialism of bodies and languages, but rather constitutes a sort of supplement to it.7 Truth, the universal, is for Badiou the exception to the situation and its state, that which “escapes” the predication of the aforementioned bodies and languages, that which escapes knowledge. In Lacan, on the other hand, the exception is an exception to the universal, rather than the universal itself.

The allegation might then be that Hegel is generative or intrinsic (the logic of contradiction) whereas formal ontology needs to be subtractive or extrinsic (the logic of exception). To respond to this critique, we need to first provide an interpretation of Hegel which allows for what is really at stake in the logic of exception, namely non-relation. Hegel is supposed to be a philosopher of pure mediation, but need he be?

First, we must indicate what we mean by relation. Relation is for Hegel mediation, a particularly intimate form of the more general category of “relation”, which includes mutual determination of identity, an interpenetration of categories. When we say that category A is dependent on its reverse, ~A (read “not-A”), we are declaring mediation. The dialectic proceeds in three (or four) steps: identity, difference (mediation), and identity of identity and difference. So we go from A to ~A (a proper contradiction) and then to A', a new category which contains both A and ~A as its inner determinations. This is the going-out into other, from A to ~A, and then a return to immediacy in A'.8

I claim that dialectical derivation or implication is Hegel's moment of non-relation. As a properly logical system (albeit a non-standard one), the logic of contradiction proceeds in the subtraction from the given field of relations represented by the categories in the current iteration of positing. Semi-formally (as in, not wholly formalized or worked out in its implications), we can write:

- A positing A
- ~A applying A → ~A
- A & ~A by adjunction
- A' applying A & ~A → A' (for some A' is)

So we have a movement, from non-relation to a filling-out of relations, in other words from immediacy to mediation (and back). Speculative reason, far from being a relation between categories, is actually their absolute non-relation. This may be somewhat controversial, since usually a speculative statement par excellence such as “Spirit is a bone” would imply that there is a hidden relation between mind and matter, the highest and the lowest. This is true, but we must tread very carefully here. “Spirit is a bone” posits an identity, not a relation. This identity, to be made explicitly true, must of course be relationalized or mediated. But this is a subsequent necessity. Speculative reason first makes the leap, but the gap over which it leaped must then be filled in with relation: “Links of this chain [of categories] are the individual sciences [of logic, nature and spirit], each of which has an antecedent and a successor—or, expressed more accurately, has only the antecedent and indicates its successor in its conclusion” (Hegel 842). Indication is the logical element, which is what connects the links of the chain in a non-relational way, i.e. in a formal way. Relation comes next.

If we consider the

*Entschluss*which opens Hegel's Science of Logic as a first exercise of non-relation (which is not such an odd thing to do considering the non-relational or singular status of the decision in Badiou), then we come to the conclusion that the very act of positing is itself non-relational. To posit is to cut off, to subtract, to reduce external relation, mediation. It is to put this one thing in front of the mind and not anything else. It is to apprehend in a rather Cartesian sense.

We can now give a Hegelian response to Badiou's criticisms as set out in the previous section. The illicit nomination, the decision which produces the transition of quantitative to qualitative (good) infinity is just a particular instance of non-relation, speculative positing, which Badiou registers more keenly than the other derivations. But it must be understood that all dialectical derivations necessitate this moment of decision/non-relation. Badiou's critique, however, also includes the allegation, necessitated by Badiou's own theory of decision, that the resulting category (the qualitative infinite) is insufficiently based on the previous category (the qualitative identity). This being-based-on can be further specified as logical implication. In other words, Badiou denies that the qualitative infinite is the consequent of a formal conditional of which the quantitative infinite is the antecedent.

We are here confronted with the question of how one category really “follows” from another. As decision, it cannot follow from the tracing of relations, cannot be a form of mediation, but nonetheless it must be formally or logically implicit. From Badiou, we learn that decision is not arbitrary in what it makes possible, or rather that there is a particular necessity to the form of the truth which decision faithfully constructs. A break from relation does not in turn imply a break from implication. The contradiction of categories A and ~A must be understood to formally imply category A', and to avoid explosion (that from a contradiction anything at all may be derived), this A' must be defined in terms of A. We cannot, in other words, derive an unrelated B or else we are not really doing a derivation at all but are immersed in formal triviality.

A' must contain the contradiction of A and ~A but must surpass them in some form. In other words, it must be both A & ~A and neither A nor ~A. If A represents a category, and ~A represents that categories “opposite” (or contradictory), then A' must include the category, the opposite, and what is in some sense “in between” those two categories, what escapes those categories, or in other words what is not covered by the relation between them. This necessitates the rejection of the Law of Excluded Middle, or whatever analogue of it that might make sense in our semi-formal representation: The truth is neither A nor ~A. It just so happens that in dialectical logic the truth also must contain the opposition of A and ~A, though now in immediate form.9 At this moment, I do not know precisely how to formalize a definition of A', or precisely how that definition would relate to dialectical entailment.

All this is certainly not to say that dialectical deduction is completely “formalizable” if by this we mean mechanizable. While we can represent the derivations in a semi-formal or formal system, this is only one aspect of dialectical development, namely the non-relational aspect. The formal structure of deduction sits atop the relationalization of the categories, though in that relationalization the structure is also “absorbed”, made intelligible. This is most obvious in the Absolute Idea, a category which might not be formalizable as a result or logical consequent. The content of the derivation must be determined, though of course only after the formal moment, through relationalization, mediation. But in the Absolute Idea, we encounter the limit of formalization, internal to its own development. This is just to say that the dialectic cannot be further dialecticized, though it can through the Absolute Idea come to “apply to” other subject matters. This “application” would probably constitute another type of decision, though one which at this time must remain opaque.

One way to understand the relation between Hegel and Badiou is to see Hegel's dialectical deduction, the moment of speculative positing, as a Badiouian truth-procedure in miniature.10 Not only is the first Entschluss at the opening of Science of Logic a sort of Badiouian decision, but each and every exercise of speculative reason is as well. Hegel then saturates, builds the truth of the previous categories by relationalizing the connection. The further derivation comes from the situation constituted by those relations, just as in Badiou. It may be that dialectical logic is generative, but then in that case Badiou would be as well. The question would be whether there is something which is decisional which is impossible to relationalize. For Badiou, that is truth. For Hegel, that is the dialectic itself, the Absolute Idea.

But Hegel also provides us with a way of answering Badiou's second critique, that contained in Logics of Worlds. Badiou alleges that Hegel's philosophy necessitates the so-called Axiom of the Whole. Hegel's “Whole” is the mythical fully-relationalized reality, one for which a further exercise of speculative positing was not possible. Within the Science of Logic, there is an analogue of this idea, namely the aforementioned Absolute Idea. The Absolute Idea is the fully explicit relationalization of what during the deductions appeared as a cyclic activity of positing and relationalizing. Can the Absolute Idea produce a dialectical successor of the sort employed throughout Science of Logic? As the completion of formalization, I claim that the only possible further “development” of the Absolute Idea is the moment of content, or perhaps rather the moment of “application”. Once we reach the Absolute Idea, we explicitly posit the positing principle (along with the other principles of dialectical deduction as referenced above). Thereafter, we may posit various objects and see where they take us, whether to the philosophy of nature, spirit, history, etc. The Science of Logic only utilizes the “Whole” to the degree that mediation is ever-expanding. But that expanding mediation is also counteracted by the final category, the Absolute Idea. The Absolute Idea is emphatically not the Whole, only the logic which “tends towards the Whole”, i.e. which expands the scope of possible mediations. The expansion, Hegel writes, “may be regarded as the moment of content” (840). But this is not the converting of form into content, but the making-intelligible of formal decision. The Absolute Idea is, explicitly and formally, the positing principle, the principle of decision and its implications. It is the summation of what came before not only in its micro-level derivations but in its trajectory.

Unfortunately for Badiou, this is not the “Whole” of which he speaks, since the positing principle constitutes a break just as much as it opens up new possibilities for mediation. I claim that we can interpret Hegel as a formal ontologist to the extent that dialectical logic is understood as the dialectic of intelligibility, of form and content. There is an end, but this end is not the Whole, rather it is the principle of the non-whole, the positing principle which disrupts the relations of what was once a sort of whole, the relational field of previously-posited concepts. Just as much as dialectical logic cannot end with the “Whole”, it cannot continue indefinitely: “It has been shown a number of times that the infinite progress as such belongs to reflection that is without the Notion; the absolute method, which has the Notion for its soul and content, cannot lead to that” (Hegel 839). This is most likely the most difficult aspect of the project we have undertaken, which certainly cannot be answered here.

Contradiction and exception are thus not necessarily mutually exclusive logics. We might be able to represent this by considering a formal deductive system in which the Law of Non-Contradiction is rejected as well as the Law of Excluded Middle (or whatever analogues could be found, if the logic is weird enough). Certainly, the logic would be very odd, and many of the old definitions and rules of classical logic would not apply. But a system may have both moments, and this is Hegel's reliance on further positings being “neither” of the previous categories but also “both” of them. While the project of formalizing such a logic is to my mind still a long way off, such a project is certainly a useful one for formal ontology, and Hegel's logic can be studied not as a one-sided decision for contradiction over exception, but an informal attempt to use both. We might then say that dialectical logic is the unity of exception and contradiction, non-relation and relation.

**V. Conclusion**

In the course of this paper, we have shown Hegel to uphold the Parmenidean thesis of the identity of thinking and being so beloved by Badiou – this is his insistence on what we have been calling intelligibility. We then considered several of Badiou's criticisms of Hegel's system, in particular as they relate to the decision, logical deduction, and the Axiom of the Whole. We then determined Hegel's logic to contain both the non-relational speculative break and the mediation necessary for subsequent speculations. However, the major work of formalization remains to be completed. Hegel, it must be admitted, was a formal ontologist before the birth of either modern formal logic or axiomatic mathematics.

The key lies in the interpretation of internality and externality in terms of formal logic: what is “implicit” in a deduction is just a true statement implied by what has been posited, but which has not itself been explicitly posited yet. It is not a separation once more of form and content, since its content is that from which we derive the logical system, the absolute form. The two moments have however been separated, namely those of relation and non-relation. But those moments are each necessary to the method of deduction, for it is out of their interplay that new categories emerge. This method of interpreting Hegel as a logician in the context of formal ontology would, however, need to be applied to subsequent other Hegelian projects, such as the Phenomenology or the philosophies of nature or spirit, to see whether the interpretation holds up in a broader context.

Works Cited

Badiou, Alain. Being and Event. Trans. Oliver Feltham. New York: Continuum, 2005. Print.

---. Logics of Worlds: Being and Event II. Trans. Alberto Toscano. New York: Continuum, 2009. Print.

Birchall, B.C. “On Hegel's Critique of Formal Logic.” Clio 9 (1980): 283. ProQuest. Web. 9 Dec. 2015.

Hegel, Georg Wilhelm. Hegel's Science of Logic. London: G. Allen & Unwin; 1976. Print.

Kosok, Michael. “The Formalization of Hegel's Dialectical Logic: Its Formal Structure, Logical Interpretation and Intuitive Foundation.” Web. 9 Dec. 2015. <http://www.thenewdialectics.org/hegeldiallog.htm>

Lacan, Jacques. The Seminar of Jacques Lacan, Book XX: On Feminine Sexuality, The Limits of Love and Knowledge, 1972-1973. Ed. Jacques-Alain Miller. Trans. Bruce Fink. New York: W. W. Norton & Company, Inc, 1999. Print.

Žižek, Slavoj. Less Than Nothing: Hegel and the Shadow of Dialectical Materialism. New York: Verso, 2012. Print.

Notes

1It is unclear, however, whether Badiou himself is a dialectician in any normal sense of the term. For claims that his thought is in fact dialectical, see Bruno Bosteels' Badiou and Politics. For claims that his thought is primarily anti-dialectical or non-relational (which is perhaps the dominant position), see Nick Hewlett's “Politics as Thought? The Paradoxes of Alain Badiou's Theory of Politics”, as well as various criticisms made by Peter Hallward in Badiou: A Subject to Truth, among other places.

2I will not go into the actual demonstration of this development, since Hegel has already done so sufficiently.

3In particular, see Seminar XX, Lecture VII.

4One the one hand, axiomatic mathematics was not around during Hegel's time, and hence the mathematics he knew did not yet create its own “object”, was not yet fully intelligible. Notably, neither was the logic of his day. Dialectical logic is probably a response to this deficiency.

5This argument is found in Being and Event, 163-169.

6It is at this point unimportant whether the Whole is ever reached, whether the dialectic is ever “completed”.

7See, for example, the opening of Logics of Worlds.

8In “The Formalization of Hegel's Dialectical Logic” (http://www.thenewdialectics.org/hegeldiallog.htm), Michael Kosok proposes a formalization of dialectical logic based on intuitionism, including an ascending hierarchy of ever-more-complicated relations and the denial of the Law of Excluded Middle. I do not have space to analyze this formalization, but it does suffer from several defects, in that it does not affirm explicit contradictions of the form A & ~A. On the other hand, his formalization is otherwise extremely attractive, and its use of presence and absence in conjunction with affirmation and negation may deal with the issue of contradiction quite well. Further, Kosok's logic does not have a conceivable halting point. At least in the Science of Logic, Hegel finally derives the explicit form of the dialectic itself. It is not wholly clear to me whether Kosok's formalization, or any pure formalization for that matter, could account for this sort of complexity. In any case, it seems to me that no Hegelian logic can be truth-functional in a traditional sense, since each category is both true and false, whether its A or ~A or A & ~A. Doing these operations, using these logical connectives, does not seem to affect the further categories. Dialectical deduction is truth-functionally useless, unless we introduce an intricate temporality to the logic, such that A' is more true (and less false? It is unclear) than its previous determinants, in an ever-ascending spiral. This is more or less Kosok's claim, and why his logic is tiered the way it is.

9Kosok creates a tiered logic for this reason, such that each higher-level category contains within itself (or is defined by) the lower-level categories. At each subsequent level, there are ever more oppositions which are internalized. But while Kosok rejects the Law of Excluded Middle, he refuses to deal with explicit instances of contradiction. See “The Formalization of Hegel's Dialectical Logic”, found at the following link: http://www.thenewdialectics.org/hegeldiallog.htm

10Credits to Frank Ruda for suggesting the connection.

[This is a paper I wrote for Fred Jameson's course on Hegel's Science of Logic. I apologize for the formatting of these academic papers but I don't have enough motivation to make it work in Blogger.]

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