On Hegel's Doctrine of Contradiction 1

Michael Wolff Universitdt Bielefeld

Translated by Erin Flynn, SUNY Stony Brook Kenneth R. Westphal, University of New Hampshire

. I .

Hegel's texts are famous for the factthatthey use the expression "contradi-ction" in quite unusual ways: this expression is almost never used for designating a specific kind of (or relation between) sentences, statements, judgments, predications, or any other linguistic constructions. Instead, Hegel uses the expression "contradiction" to designate something objective, some-thing about the things about which we speak and judge.

Hegel's usage has been assessed variously. Some take it as a sign of his logical illiteracy; they reproach Hegel for unwittingly transferring a logical concept to extralogical objects. Others see in precisely this transfer an at-tempt to erect a "higher" form of logic that repeals the laws of ordinary elementary logic. Finally, there is also a tendency to interpret Hegelian "contradiction," whatever it might mean, as a nonlogical concept in order to extricate Hegel's philosophical logic from conflict with conventional elementary logic.

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Now all of these assessments rest implicitly on a certain preconcep-tion of what exactly a contradiction in the conventional logical sense of the word is. Thus there is a question whether this preconception about contradiction's essence already differs from Hegel's; whether, therefore, Hegel's particular usage is perhaps based on a distinctive view of the essence of contradiction.

This question should not be answered without looking at the texts in which Hegel addresses the question of what a contradiction actually is. The section of the Science of Logic titled "Contradiction" would appear to contain a systematic exposition of the concept, and I believe that for our question this section deserves central attention.2 Most important is the fact that Hegel in no way limits the object of his conceptual exposition to any specially conceived kind of contradiction; Hegel seeks to explain the very essence of contradiction. Thus he appears not at all to acknowledge something like a particular "dialectical" contradiction, certainly not any "dialectical" contradiction that contrasts with any specifically "logical" contradiction. Perhaps the one place Hegel's texts expressly mention dia-lectical contradiction (in the Vorlesungen aber die Asthetik3) implies no difference between "logical" and "dialectical" contradiction, but can be interpreted as saying that, on Hegel's view, a contradiction is something dialectical, or at least it can be considered as something dialectical.

Hegel's systematic exposition of the concept of contradiction in the Science of Logic can be understood as an attempt to explain systematically the common logical concept of contradiction. Indeed, shouldn't one ex-pect that a book bearing the title Science of Logic concerns logical objects?

The considerable difficulties that the section on contradiction poses for us consist for the most part in the fact that Hegel regards contradiction as a relation belonging to what he calls "objective logic." Whatever Hegel may mean by "objective logic," his view of contradiction as an "objectively logical" relation appears to be expressed precisely in this usage, from which Hegel never deviates in this section. By this usage, a contradiction is some-thing objective, something that belongs to things themselves. "All things are in themselves contradictory" (SL, p. 439; WL II, p. 58). In this proposi-tion, from the third remark to the section on contradiction, Hegel summarizes his doctrine of the essence of contradiction.

To elucidate in detail Hegel's doctrine of contradiction as an objec-tive logical relation would require a thorough commentary on this sec-tion.4 Here I undertake instead a task prior to such a commentary: to clarify

Hegel's Doctrine of Contradiction 3

the general sense of the question that forms the background of Hegel's sec-tion: What is the essence of contradiction? To what extent does this ques-tion pose a philosophical problem for Hegel? By considering this problem can we come to understand contradiction as a relation pertaining to "objec-tive logic?"

These questions may astonish. Hasn't formal logic since its Aristote-lian beginnings provided a more or less sharp definition of the concept of contradiction? Do we still need a specifically philosophical exposition of this logical concept? Moreover, doesn't the current, sharp definition of this term by reference to logical form block any possibility of ascribing anything like "objectivity" to contradiction, without completely altering the meaning of the term? Two points in reply should be considered directly:

1. Regarding formal logic's definitions of contradiction, one should note that these have always been adapted to the purposes of a theory of formal (in particular mathematical) inference. On the basis of this orienta tion, the question of what a contradiction is has a quite special sense for formal logic: for formal logic, this question is directed from the start at the formal characteristics of a linguistic construction by which one can iden-tify the presence of a contradiction in formal languages. Only in formal languages are formal characteristics necessary and sufficient criteria of con-tradiction. (Conversely, formal languages can be defined precisely by the fact that their symbols directly and unambiguously indicate the presence of a contradiction.)

Thus, for example, the most common definition since Wittgenstein-that a contradiction is a statement that is false on (formal.) logical grounds-concerns solely contradictions in formal languages. The phrase "false on logical grounds" here of course does not mean, "false on the basis of an internal contradiction;" that would violate the common rules of defi-nition. Rather, the statement is "false on logical grounds" insofar as it is composed of partial statements that, due to their form, are related as the affirmation and negation of some statement p. However, not all statements with this contradictory form also have a contradictory content. Rather, as Kant stressed in his "transcendental dialectic,"S they can instead have a contrary or subcontrary content. Thus in nonformallanguages, a statement's form, in the sense specified by the formal-logical definition of "contradic-tion," is not a sufficient condition for the presence of a contradiction. Indeed, it is not even a necessary one. Ordinarily we regard as contradictory those statements that attribute to an object predicates that are incompatible in

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any conceivable context. Such predicates need not be mutual negations; they can also be related as "married" and "bachelor" or as "larger than the sun" and "smaller than the sun." Predications that are contradictory in this sense are counterparts to so-called analytic judgments. Again it was Kant (in his concern for a concept of contradiction not merely regarding logical form, but one suited to a "transcendental logic") who called attention to these counter-parts. Kant coined for them the term "analytic oppositions" and accounted for the concept of contradiction in terms of them: contradictions are, accord-ing to Kant, negations of analytic judgments.6

Hegel's Science of Logic is intended to be the "science" of a logic that is not merely formal. Like Kant, therefore, Hegel is concerned with a con-cept of contradiction that does not merely regard logical form. As can be easily established, however, Hegel does not account for the concept of con-tradiction by reference to analyticity. Thus he also avoids all the difficulties that this concept entails and that to this day, as Quine made clear, have not really been solved. The difficulty of saying what actually makes an analytic judgment analytic reveals itself, e.g., in the circular way in which Kant attempts to explain in turn the concept of analyticity by means of the concept of contradiction. Hence Kant's concepts of analyticity and con-tradiction have been characterized correctly by Quine as "the two sides of a single dubious coin.,,7

So although Hegel must employ other means than Kant to explain the logical concept of contradiction, he shares with Kant-this is now my thesis-the goal of laying the foundations of this concept deeper than is either possible or desired within formal logic. Hegel proceeds from the by-no-means-misguided conviction that a formal contradiction, insofar as it is merely formal, cannot be a genuine contradiction. "A contradiction," he writes in the Elements of the Philosophy of Right, "must be a contradiction with something, that is, with a content which is already fundamentally present as an established principle" (135). According to this conviction, the merely formal logic of contradiction can have no concept in the strict sense of the word; instead, it fixes certain regular representations that, in ordinary, philosophically unreflective language, we may also otherwise as-sociate with the expression "to contradict," and epitomizes these ideas in its definition of (NB: formal) contradiction.

2. Likewise, ordinary, philosophically unreflective language also oc-casionally associates the idea that contradiction is something objective with the expressions "to contradict" and "contradiction." Outside contexts of

Hegel's Doctrine of Contradiction 5

philosophical logic, by the mid-eighteenth century at the latest, this idea appears in literature in authors such as Diderot, Voltaire, Lessing, Rousseau, Forster, and Wezel.8 Isn't this, however, a merely metaphorical usage when these authors characterize, e.g., organisms, persons, or the system of civil society as "contradictory"? Thus I come to the second question: whether precisely the logical meaning of the word "contradiction" is lost if one at-tributes objectivity to contradictions and contradictoriness to things. Doesn't Hegel make his attempt to explain the concept of contradiction dubious by objectifying contradiction?

Below I shall try to make clear that the idea that (logical) contradic-tion is something objective lies at the very basis of Hegel's exposition of this concept. Before that, however, I must dispel some prevalent reserva-tions about the possibility of such a basis. These all stem from the reproach that Hegel's concept of contradiction is based on a homonym.

It is a remarkable, though rarely observed fact that in various epochs of the history of logic very different logical uses (and if one will: meanings) of "contradiction" are current. Whether this expression is used in the sense of a particular class of false statements, whether it denotes relations between statements, or statements and facts, or things and predicates, etc., or whether, finally, it is used to denote a specific act of contradicting-we may trace all of these uses back to various meanings of "contradiction." Yet the history of logic shows that all of them were suitable for characterizing one and the same logical fact. At the very least, we find that formulations of the law of noncontradiction have been guided now by this, now by that usage. Today, for example, common formulations are based on the concept of contradic-tion as a composite, logically false statement: the law thus means that a contradiction, i.e., the assertion of a statement p together with its negation, is always false. In Plato's Republic (436b ff.), one finds the principle that it is not possible to do or to suffer opposites at the same time, at least not in one and the same respect. Aristotle takes up this "principle of action" in his Metaphysics (Book D, chapters 3 and 4), when he selects for the law of con-tradiction the formula that one cannot, if one states anything at all, state opposites, at least not in one and the same respect; otherwise one would say absolutely nothing. Contradiction is for Aristotle an act; self-contradiction is the special case of an empty action. Kant departs from Aristotle when he gives yet another formula, asserting in the Critique of Pure Reason that the law of contradiction means "that no predicate contradictory of a thing can belong to it" (AI 51jB190). Evidently Kant follows a paradigmatic usage

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different from Aristotle's; evidently he has in mind the contradiction that holds between a statement (a predication) and a fact (the characteristic [Bestimmtheitl of an object). It can be shown that Kant's own usage comports with his defining the concept of contradiction by reference to the concept of analyticity (see below). For the moment, however, only the following is im-portant: whatever objections one may have to older logical expressions, they are not based on homonyms. These modes of expression do not simply differ in meaning, they are semantically related; they are not homonymic, but paronymic.

I propose to interpret Hegel's usage also as paronymic. However, the paronym involved in Hegel's concept of contradiction represents an alto-gether new variant in the history of logic. Hegel's "contradiction" is neither a statement nor an action, neither a relation between statements nor a rela-tion between statements and the characteristics of things. It is rather, in Hegelian terminology (to be explained below) a relation between the "deter-minations" (Bestimmungen) and the "determinatenesses" (Bestimmtheiten) of an object. An example may give a preliminary illustration of this relation: when Hegel, in a remark to his section on contradiction, characterizes move-ment as an "extant contradiction," this does not mean that the movement of a certain object contradicts any statements about that object_ It means that at a specific point in time an object, insofar as it moves, occupies both the location within a specific place and the location outside that place: change of place consists in the contradiction of the locations that a moved object occu-pies. Moreover, the paronym present here appears clearly in the fact that the "existence" of contradiction, on Hegel's view, opposes the conventional law of noncontradiction: this law excludes the very possibility of both ascribing and denying a specific location at a specific point in time to an object. In Hegel's view, however, formal logic (since it doesn't even possess a concept of contradiction in the strict sense of the word) may not interfere with the affairs of an "objective logic." On the contrary: the formal validity of the law of noncontradiction (which Hegel also acknowledges9) should instead be explained on the basis of a properly developed concept of contradiction. According to Hegel, the validity of this law rests not on the fact that it would be false to attribute to things contradictory determinations and determinatenesses, but on the fact that objective contradiction exists only as "self-dissolving" contradiction. Once again this idea can be clarified, provi-sionally, by example: the contradiction of locations occupied by a moving object {insofar as it moves} cannot persist for any length of time, however

Hegel's Doctrine of Contradiction 7

brief; the movement of the object precludes its occupying anyone of its locations for any length of time, however brief.10 That the objective contra-diction "dissolves itself" is of course an idea completely foreign to formal logic. However, from Hegel's point of view, the formal-logical law of noncon-tradiction is nothing but a pale imitation of this idea, projected onto the level of linguistic rules or "laws of thought."

Paronymic shifts in meaning are an important and widespread aid in all scientific and philosophical artificial languages. Science and philosophy are for the most part fairly innovative in this respect. As is well known, words such as "warmth," "mass," "force," "weight," or "water"-words that are also used outside of physics and chemistry and are older than these sciences-gain through physics and chemistry entirely new meanings that, nevertheless, have content in common with prescientific usage and as such relate paronymically to them. Because of the limited scope and precision of expressions in natural language, paronyms prove to be useful for repre-senting linguistically newly discovered scientific domains. What we mean by the original words rooted in natural language is often made comprehen-sible to us only through their artificial, paronymically constructed derivatives. What holds of science holds too of philosophy. "Philosophy," writes Hegel in the Science of Logic, "has the right to select from the language of common life, which is made for the world of representational thinking, such expres-sions as seem to approximate to the determinations of the concept. There is no question of demonstrating for a word selected from the language of common life that in common life, too, one associates with it the same concept for which philosophy employs it; for common life has no concepts, but only representations, and philosophy itself consists in recognizing the concept of what is otherwise mere representation" (SL, p. 708; WL II, p. 357).

- II -

I hope that these considerations of paronyms and the relation between Hegel's undertaking and the formal-logical definition of contradiction dis-pel some of the common reservations about the very possibility of Hegel's project. I also believe that without these considerations no clear insight into the sense of Hegel's dialectic is even possible, for the doctrine of con-tradiction constitutes its very core.

The foregoing considerations can at most only make plausible that Hegel's question about the essence of contradiction makes good sense and

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that a satisfactory answer to it does not preclude the objectification of contra-diction. In what follows I would like to show what insights led Hegel to transcend Kant's concept of contradiction and to construct the concept of objective contradiction.

Adumbrations of these insights are contained in Kant's philosophy itself, more precisely in the theory that may be called Kant's doctrine of opposition. In addition to the concept of analytic opposition (mentioned above), Kant's philosophy contains two other concepts of opposition: "dia-lectical" and "real" opposition. I would like to show that Kant's theory of dialectical opposition represents in a certain respect a predecessor of Hegel's doctrine of contradiction: it removes the basis of the traditional way of distinguishing (common in traditional scholastic logic) between so-called "contrary" and "contradictory concepts." We must now see how Hegel completed this overcoming, begun in Kant's dialectic, of a prejudice in traditional logic.

Furthermore, I shall show that Kant's theory of real opposition pre-saged the concept of negativity that is basic to Hegel's doctrine of contra-diction. What unites both theories is the fact that they both proceed from the insight that {arithmetic} negativity is not reducible to (logical) nega-tion. This relatively new insight (which mathematics only gained in the second half of the eighteenth century) was no less fundamental for the emergence of Hegel's dialectic than it was for the emergence of modern mathematical logic (in particular, of Boolean algebra, developed soon after Hegel's death). While it turned out that, for mathematical-technical reasons, the algebraicization of logic required a completely different cal-culus from that of (traditional) mathematical algebra, philosophers such as Kant and Hegel were led to explicate the difference between negation and negativity conceptually.

In one sentence, my point in what follows is to show that, contrary to a criticism of Hegel popular from the nineteenth century to today,11 Hegel's doctrine of contradiction does not result from any unclarity about the dif-ference between real, dialectical, and analytic opposition made clear by Kant; on the contrary, it results from Hegel's critical assessment of this difference. Before examining the essential features of Hegel's critique of Kant, we must consider Kant's views more closely.

Kant's doctrine of opposition12 admits the name "contradiction" only for one of the three types of opposition just mentioned, namely, analytic

Hegel's Doctrine of Contradiction 9

oppositIOn. Specifically, this is a determinate relation between a thing and a predicate that is predicated of the thing. Kant calls this relation "ana lytic" for the following reason. On Kant's view, the predicates of affirma tive or negative subjectpredicate.judgments attribute to a thing either a so-called determination or a lack of determination. In Kant's terminol-ogy: a subject-predicate-judgment "posits" (setzt) a determination of a thing, or it "sublates" (aufhebt) it. Kant also calls determinations "real predicates"--in contrast to "logical predicates," as he calls predicates of judgment. In contrast to these, determinations of a thing are not components of a judgment or a statement; instead, they are that which the thing is, if the judgment or the statement is true. Because this truth is not fixed inde-pendently of what the thing is, one should say more precisely: determina-tions are that which the thing ought to be according to judgments that presume to be true. Therefore, if we speak of determinations or of lack of determinations, we are already dealing not with the objects of formal logic, but with the intended objects of our judgments and statements; more precisely, with the objective correlates of logical predicates. Ac-cording to Hegel's systematic arrangement (based on the Kantian termi-nology just mentioned), this is no longer the realm of "subjective," but of "objective logic." According to Kant, there is a relation of contradiction between thing and predicate only due to the fact that the thing, as a subject of (acts of) predication, is already determined by determinations or by a determinate lack of determinations. And indeed, this determi-nateness results from the fact that certain analytic subject-predicate judg-ments hold of that thing. The contradiction consists in the fact that the predicate "posits" or "sublates" a determination that the thing, accord-ing to an analytic judgment, either lacks or possesses.

I have summarized Kant's concept of analytical opposition (which of course requires more thorough examination) again in order to explain some terminological points that are significant for the other two concepts of opposition, as well as for Hegel's reflections. Consider now the other kinds of opposition.

These are oppositions without contradiction. However, it is characteris-tic of dialectical opposition that it is not a real, though it is indeed an apparent contradiction. For example, according to Kant, Zeno's paradoxes of motion are apparent contradictions; these provided the model for Hegel's concept of motion as "extant contradiction." Unfortunately, Kant provides no satisfac-tory justification for the claim that the contradiction in these paradoxes is

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merely apparent. However, in his assessment of these paradoxes Kant makes an interesting observation that Hegel developed in his logic. Kant contends that the contradiction between judgments such as 'an object x moves' and 'x does not move' resolves itself as a merely apparent contradiction when one presupposes that the object x neither moves nor does not move (AS02-3/ BS30-l). If, for example, the object x is not Zeno's flying arrow, but is in-stead the universe, then those formally contradictory judgments both prove to be false, and therefore do not relate in a truly contradictory sense; for the universe, thought to be the totality of all things, can neither change nor maintain its place, since there is no such place. In the case of the universe we presuppose an object of which neither of the contradictory predicates may be predicated. What may be predicated of it are only the negations of these contradictory predicates.

At first glance, Kant's position appears to violate the law of excluded middle: if both of two contradictory predications are negated, then there is a new contradiction. However, Kant notes, this violation in fact does not occur in all cases. Kant offers a trivial example: it may be maintained that 'a body smells good' and 'it does not smell good.' Formally, this is indeed a contradiction. However, the content of these judgments shows that the contradiction is merely apparent, if both claims are negated and if the sub-ject term denotes an object that does not and cannot even have an odor, and therefore smells neither good, nor not good. Despite their contradic-tory form, such predications thus prove to be mere contraries. The converse case also holds: two formally contrary predications can be genuine contra-dictory predications, if the presupposed meaning content of the subject term-in other words, if the presupposed determinateness of the object-is changed.

Kant here articulates, albeit indirectly, a paradigm case of an elemen-tary law of a special kind of logic that, following Hegel, can be called the "logic of reflection.,,13 Kant articulates a logical law that is not a law of logical form. This logical law of reflection says: given a constant logical form of two predications, the relation of contrariety can be transformed into the relation of contradiction, and vice versa, if only the presupposed determinateness of the object is changed. This logical law of reflection can also be put this way: given the meaning of a pair of predicates, if it is pre-supposed that these predicates, when referred to the same subject term, result in logically contradictory predications, then this presupposition also contains some material presupposition regarding the determinateness of the

Hegel's Doctrine of Contradiction 11

object (of that subject term). In short, whether there is a genuine (and not merely a formal logical) contradiction between two predications depends on the presupposed determinateness of the object. I shall call this presupposed determinateness of the object the" substrate of logical reflection." Whether two predications with opposed contents and logical forms result in a genu-ine contradiction thus depends upon the substrate of logical reflection.14 Now according to Kant, not all apparent contradictions between subject-predicate judgments are dialectical oppositions. They are only dialectically opposed if their contradictoriness depends upon a special substrate oflogical reflection, one that has a nonempirical, transcendental character. The con-tradictoriness of dialectically opposed judgments depends namely on our tacitly taking the things judged for "things in themselves." Kant's so-called Antinomies are based on dialectical oppositions in this strict sense. They are pairs of subject-predicate judgments, contradictory with regard to logical form, that contain a special, common, unfulfilled presupposition concern-ing the object of the judgment: the presupposition that it concerns a thing in itself. If this presupposition is cancelled [aufgehobenl, then both judgments, regarding their content (without change in logical form), change from con-tradictory to contrary or subcontrary judgments.

Although Hegel explained the details of Kant's Antinomies quite dif-ferently than Kant, Hegel nevertheless held that the discovery that the contradictoriness of the antinomical judgment pairs depends on the pre-supposed content of the subject terms of these judgments was one of Kant's great achievements, as Hegel recalls once again in the famous section on method at the end of the larger Logic (SL, pp. 832-34; WL II, p. 494).

It is important to note that Hegel himself reassessed this discovery by using conceptual means that Kant developed in his analysis of the third form of opposition.

The third form of opposition, real opposition, has even less to do with logical contradiction than does dialectical opposition, which at least is an apparent contradiction. In real oppositions, formal negations do not even occur. At issue are pairs of predicates of exclusively affirmative subject. predicate judgments that have oppositions as their content, despite their affirmative character. However, the opposition results not from the logical negation of predicative judgments, but rather from the negativity of their basic "real" predicates, that is, of the determinations of the object of judg-ment. The introduction of the concept of negativity in Kant's precritical "Versuch den Begriff der negativen Groj3en in die Weltweisheit einzufuhren"

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(1763),15 was of great (though little understood) significance for Kant's later philosophy, and also for post-Kantian, Hegelian, and materialist dialectic. The expression "negativity," as the substantive of the adjective "negative," apparently only became a philosophical term of art after Kant. What is being negative (Negativsein)? What is negativity?

One must bear in mind that for Kant, negativity is first of all a math-ematical concept, i.e., a concept borrowed from mathematics that has noth-ing to do with the various logical forms of inner and outer negation, or indeed with declarative or predicative logical functions. Hegel's contempo-rary interpreters would be well advised to consider the content of the math-ematical concept of the negative in interpreting the concept of negativity, instead of as Dieter Henrich recently suggested, directly tracing this concept to the substantivization of the negative form of assertion.16 As far as Kant is concerned, he owes his concept of negativity to a differentiation between "negative" and "positive quantities" that became possible in the eighteenth century as a result of the Newtonian revolution. Newton himself first advo-cated using the mathematical concepts "affirmative" and "negative quanti-ties" in natural philosophy in his Optics. This was quite unconventional and offended contemporaneous logicians. Logicians (among them Christian Au-gust Crusius, whom Kant criticized on this count) declared Newton's interpretation of repulsion as a negative quantity to be absurd because they misunderstood the concept of negative quantity simply as the concept of a logically negated quantity. However, Newton's intention was not the victim of mere superficiality. This very misunderstanding showed at once that, up to then, no one possessed an unambiguous, consistent concept of the negative-not even mathematicians in their foundations of arithmetic and algebraic operations.

In his history of algebra, 1 7 Florian Cajori recounts the paradoxes in which even great mathematicians, such as Leonard Euler, entangled them-selves in developing the rules of signs. There was no clear concept of the difference between negative numbers and subtracted numbers in mathemat-ics until the nineteenth century.

Kant, who follows Kastner,18 is also not clear about this difference, and hence achieved no satisfactory concept of negativity. Kant interprets numbers that are less than zero as quantities that are less than nothing, and that therefore cannot exist. Nevertheless, he attempts with the means at his disposal to develop a consistent concept of the negative in a nonlogi-cal sense (by which he aims to rescue Newton's great program in natural

Hegel's Doctrine of Contradiction 13

philosophy). Briefly, Kant's result is this: negativity shares with both logi-cal negation and contradiction the fact that it results in a determinate sublation (Aufheben). However, what is sublated (aufgehoben) are not the determinations that truly belong to a thing according to an analytical judg-ment. Therefore, the thing as a subject of predication is also not sublated (hence it is not a nihil negativum). Instead, only "consequences" of the op-posed determinations mutually sublate themselves. These consequences do not become "nothing," but they can equal zero. For example, if something moved by a force is moved at the same time by an equivalent force in the opposite direction, this results, not in one of the two physical effects (a motion), but in rest (that is, to a motion=O). Or, as Kant puts it, if someone has a capital worth of $100 dollars and at the same time a passive debt of $100 , then the consequences of these means (e.g., income and expenses) mutually cancel (aufheben), and the result is capital, not equal to $100, but equal to O.

Kant gives a kind of dynamic explanation of mathematical negativity. Negativity is a relation of "capacities," "forces," or "causes" that mutually deprive each other of their "consequences" or "effects." Kant calls this kind of deprivation, in contrast to logical negation, "privation." It is negation in a nonlogical sense.

Although Kant's dynamic, privative interpretation of negativity exhib-its serious mathematical deficiencies (as can be easily proven), it is to his credit that he made the concept of negative determination fruitful outside of mathematics, and most of all for the aims of criticizing metaphysics. Pre-Kantian, especially Leibnizian-Wolffian metaphysics assumed that the logical predicates expressible in two contradictory judgments do not both have a determination as their content, but that one of the two always has an absolute lack of determination. All possible determinations taken to-gether constitute the so-called sum total (Inbegriff) of reality, while each lack of determination is a limit of reality. "Reality" was conceived of as a great metaphysical cake, so to speak, of which things, as bearers of determi-nation, get larger or smaller portions. The limitations of these portions are the limits of reality. All true negations predicated of these things have these limits, an absolute not-being (Nichtsein), as their content. With the conceptual resources of his theory of negativity Kant began his assault on this Leibnizian-Wolffian metaphysical cake. Already the precritical Kant, though especially the author of the Critique of Pure Reason,19 pointed out that real negations need not contain limitations of reality; they could instead

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mean privations. From Hegel's point of view, however, Kant's critique of metaphysics stopped halfway. That is, Kant sublated (aufgehoben) the metaphysical distinction between reality and limitations of reality only for the domain of objects of appearance, but not for the domain of things in themselves. According to Kant, privations, and therefore also negative de-terminations, do not belong to things themselves; what pertains to them instead is reality in the sense of the "transcendental ideal," or negation in the sense of an absolutely transcendental not-being. For all empirical deter-minations, insofar as they do not belong to things in themselves, must in truth be negated as determinations of those things in themselves. The "tran-scendental ideal" is, so to speak, the metaphysical cake risen (aufgehoben) in Kant's critique of metaphysics, preserved (aufgehoben) in the pantry of things in themselves.

- III -

Hegel extended Kant's critique of metaphysics by (quite rightly) elimi-nating as groundless the distinction between the negativity of determina-tions and transcendental negation. Hegel shows, with good reason, that negativity (whether within or outside mathematics) is not at all based upon the mutual privation of opposed capacities, forces, or grounds, but can be explained by the fact that two determinations that differ in content stand in relation to a substrate of logical reflection and become, by virtue of this relation, "opposed" determinations. Hegel rejects the dynamic interpreta-tion of negativity.

This idea, which Hegel develops in the section "Opposition" (SL, pp. 424-31; WL II, pp. 44-48-directly preceding the section on contradiction), says the following: A determination is negative in relation to another, not insofar as it dynamically deprives the consequences of the other, but rather, insofar as it is based on a substrate of logical reflection, due to which the other determination relates to it as its opposite determination. Hegel thus merely draws a consequence of the law oflogical reflection mentioned above. As we saw, Kant ascertained that the contradictory relation of two predica-tions depends upon the kind of substrate of logical reflection to which they are related. Two predications relate as contradictories if and only if each of these predications is equivalent to the logical negation of the other. It fol-lows from this logical fact that this equivalence likewise depends upon the relation to a substrate of logical reflection. It also follows that, of two

Hegel's Doctrine of Contradiction 15

contradictory predications, both have negating as well as determining char-acter: neither indicates an absolute lack of determination; instead, both attribute a determination to the object, though each of these determinations is the negative of the other-

Like Kant, Hegel also claims that his reflections on the concept of negativity provide a conceptual clarification for elementary mathematics. In a remark to the section on "Opposition" (in the large Logic) in which he-likely inspired by the mathematician and student of Kastner, Kliigel20-pro-vides a more exact analysis of the different meanings of opposed signs for algebraic values and operations, Hegel actually anticipates an important idea in the newer algebra and vector algebra. Since Hermann GraBmann, who, incidentally, studied Hegel's Science of Logic, mathematics has had the concept of absolute value, through which the negativity of a number or a vector can be defined consistently_ 21 Two opposite numbers or vectors, +a and - a, are based on the same absolute value I a I . This basic absolute value is, accordingly, something like a special case of a substrate of logical reflec-tion. It is the "bearer" of two quantitative determinations, +a and -a, that one must presuppose in order to regard both determinations as opposite, that is, as positive and negative. These determinations can be called "oppo-site" insofar as the positively determined value is exactly that value that is not negatively determined, and vice versa. In the remark to his section on opposition, Hegel provides a careful and path-breaking analysis of the alge-braic concept of negativity and-despite his peculiar terminology and (by contemporary standards) somewhat unhelpful symbols22-introduces the concept of absolute value as the concept of a substrate of logical reflection_ With much more right than Kant, Hegel claims to provide the conceptual means for solving the algebraic paradoxes.

Now, for Hegel, just as for Kant, negativity is not only a mathemati-cal concept_ Indeed, on Hegel's view, the concept of opposition (Gegensatzes), upon which all of Kant's concepts of opposition (Oppositionsbegriffen) are based, cannot be explicated at all without the concept of negativity. On Hegel's view, each opposition is a relation of something positive and something nega-tive. This relation of negativity thus has a quite specific logical (not, as with Kant, dynamic) structure_ However, Hegel distinguishes different "forms of the positive and negative" according to different forms of arithmetic oppo-sition (SL, p. 428; WL II, p. 45)- To understand Hegel's exposition of the concept of contradiction, one must grasp Hegel's reason for these distinc-tions in form and his analysis of the logical structure of negativity.

16 The Owl of Minerva 31: 1 (Fa111999)

Concerning the logical structure of negativity, Hegel characterizes it as "second negation," in order to distinguish it from so-called "simple" logical negation. This structure can thus be described as a particular relation of identity: X and Yare negatively related if and only if X is identical with not-Yand Y is identical with not-X. If X and Yare two (positive) determina-tions (e.g., two colors, such as blue and yellow; two vectors, such as six miles from east to west and six miles from west to east; or two sums of $1,000 capital and $1,000 debt), and if we say that these determinations are op-posed and are related negatively (in the sense specified), then this negativity holds only under a quite specific presupposition. Blue is not simply identical with not-yellow (for green, colorless, etc. are also not-yellow). The same holds for the other X and Y: only as determinations of a determinate object, that is, only in relation to a substrate of logical reflection, are X and Y related negatively. Hence the supposition that blue and yellow are related negatively presupposes a quite specific theory of opposite colors (e.g., Goethe's doc-trine of colors23), according to which blue, as a color of a certain type (Goethe and Hegel would say, a "simple complementary color"), is identical to not-yellow. Likewise, a six-mile stretch from east to west is identical with the stretch not leading six miles from west to east, only insofar as it is a distance with a quite specific vector value and a quite determinate location. Finally, it only makes sense to speak of "opposed," "positive" and "negative" capital, if different capital resources of a determinate absolute value are compared with each other.

Hegel's conception of the logical structure of negativity enables him, furthermore, to endow the traditional formal logical law of excluded middle with a quite special sense. Though this law "usually," Hegel writes, means "that, of all predicates, either this particular predicate or its non-being be-longs to a thing," it ought henceforth to mean, "that every thing is an oppo-site, it is determined as either positive or negative" (SL, p. 438; WL II, pp. 56-57); whereby Hegel expressly stresses that "the opposite" does not here mean "merely lack or indeterminateness," as usually understood in formal logic as "logically opposed predicates." Hegel thus holds that of two positive determinations, X and Y, that are related negatively (which Hegel thus desig-nates, according to algebraic sign language, by the symbols" + A" and" - A"), one always belongs to something (to a thing). There is no middle, no medius terminus here. Hegel's view is evidently false if the something in question is not a completely determinate something, if, in other words, the deter-minateness of this something is not a substrate of logical reflection. And

Hegel's Doctrine of Contradiction 17

thus Hegel also ascribes to this something a determinateness that he sym-bolizes by the unsigned letter "A" (analogous to the algebraic symbol he uses for absolute value), which we can appropriately regard as the sub-strate of logical reflection, on which the relation of negativity between +A and - A depends_

Now the opposition of determinations is, according to Hegel, only one of the "forms of the positive and negative." Another "form" results from the fact that the opposite determinations as such, that is, as opposite, are op-posed to the substrates oflogical reflection: as substrates oflogical reflection, these are the nonopposed determinatenesses of those objects that are bases for opposite determinations. This opposition of the opposed and the nonopposed is, as it were, a relation of opposition on a higher level, and Hegel places great value in the observation that this elevated opposition also appears as the object of mathematics and is the basis for certain higher arithmetic operations in which calculations use positive and negative abso-lute values (SL, pp. 429-31; WL II, pp. 47-48).

Likewise, the higher relation of opposition contains the logical struc-ture of negativity: the nonopposed is identical with that which is not oppositej and vice versa, the opposite is identical with that which is not the not-opposed. This identity must also be clarified, for Hegel does not mean it in the trivial sense which initially seems to be expressed here. Insofar as "the opposite" always means (positive) determinations, and the nonopposed always means substrates oflogical reflection, the following results: the substrate oflogical reflection A is identical with that which is neither +A nor - Aj and, con-versely, what is +A or - A is identical with not- A. This identity holds, on the one hand, insofar as A is the determinateness of an object that is the basis of the determinations +A or -A, and, on the other hand, insofar as +A or -A are determinations of an object determined as A.

It is therefore not at all trivial to assert the negativity of the nonopposed and the opposite, for the nonopposed is something opposite precisely inso-far as it is identical with that which is not the opposite. It is no less paradoxical that the opposite should be something opposite as the correlate of the nonopposite. It appears to be in exactly one and the same respect that the relata of higher level opposition are related both negatively and as not-negatively.

We have now reached the genuine core idea of Hegel's section on "Contradiction." To develop it in detail requires, as mentioned, a thorough commentary. Here some key points of this idea must suffice.

18 The Owl of Minerm 31: 1 (Fall 1999)

1. That which Hegel calls "contradiction" is a relation between one of two opposed determinations and the substrate of logical reflection with regard to which the determinations are mutually opposed. The relata of the contradiction are therefore those of the higher relation of opposition just mentioned. Hegel designates these (for reasons that need not detain us here) as "self.sufficient determinations of reflection"; the substrate of logical re flection Hegel calls "the positive in itself," in opposition to the "negative in itself" opposed determination. The contradiction consists precisely in the fact that the selfsufficient determinations of reflection are related in one and the same regard as negative and also notnegative. The proximity to Kant's concept of contradiction is still easy to see. Hegel's substrate oflogical reflection replaces Kant's analytically ascribed determinateness of a thing; Kant's "logical predicates" that "contradict" this determinateness if the con tradictory opposites of these predicates analytically follow, are replaced by Hegel's "real predicates": (positive) determinations, whose opposition or negativity depends on their relation to the substrate of logical reflection.

2. Contradiction thus becomes an "objectively logical" relation. (Here this amounts to saying only that it is a relation of objective determinations and objectively determinate objects.) More precisely, contradiction becomes a relation of objective logical reflection. Eighteenthcentury logic regarded logical reflection as nothing objective, but rather as a subjective activity of the power of judgment: logical reflection consisted in identifying, distin guishing, opposing (etc.) logical predicates in regard to their content. Hegel transforms the traditional logic of reflection into an "objective logic" in two ways: first, he redirects attention from relations among logical predi cates to relations among determinations that are already presupposed by those relations among logical predicates (in identifying, distinguishing, opposing [etc.] logical predicates, we already presuppose the identity, dif ference, or opposition of determinations). Identity, difference, opposition, etc. do not interest Hegel so far as they are concepts of our subjective reflec tion (as Kant's "concepts of reflection"), but only as determinations of re lations among determinations (as "determinations of reflection"). Second, Hegel uses, as mentioned, Kant's discovery in the Dialectic of Pure Reason that logical relations such as contrariety and contradictoriness cannot hold between logical predicates alone, but rather, only insofar as these predicates are related to a substrate of logical reflection. Transferred from the logical level of predicates of judgments to the real level of objective determina tions of objects (Gegenstandsbestimmungen), this means that determinations

Hegel's Doctrine of Contradiction 19

of reflection such as diversity and opposition (which correspond on the real level to the logical relations of contrariety and contradictoriness) cannot obtain by themselves, but depend on the internal relations of determina-tions to specifically determined objects. The determinations of reflection are mirrored, so to speak, in the internal relations of individual things, just like Leibnizian monads. Their external relations of opposition are mirrored in the internal contradiction of individual things.

3. Hegel's talk of objective contradiction (something that inheres in all individual things, so far as they are mutually contradistinguished only by certain opposed determinations) entails that genuine contradictory judg-ments need not simply be false. For genuine judgments of this type are supposed to consist precisely in ascribing objective contradictions to objects. (This follows from the considerations above regarding Hegel's paronymic sense of "contradiction.") Hegel compares this ascription of contradiction with the tautology that is the basis of the law of identity; the tautology is true, and nevertheless it "says," in a certain sense, "nothing" (SL, p. 439; WL II, p. 58). The vacuity of the law of identity lies in the fact that it says of everything: it is precisely what it is. "The contradiction that appears in oppo-sition, is only the developed nothing that is contained in identity and that appears in the expression that the law of identity says nothing" (SL, p. 439; WL II, p. 58}.24 The "nothing" of contradiction consists in the fact that all individual things, due to their determinateness against each other, relate nega-tively to themselves. Negativity as self-relation both constitutes and dissolves contradiction. Negativity as self-relation is the reason why things are not simply what they are, why they cannot consist in that which they are, but, to use Hegel's words, must "perish" (zugrundegehen).

Hegel's doctrine of contradiction still remains the scandal it is taken to be. The view that genuine contradictory judgments need not be false lies out of the bounds of any ordinary logic. Still, it is difficult to find a logic on the basis of which Hegel's theory of contradiction is vulnerable. Contrary to common supposition, formal logic provides no such basis for criticism. To be sure, in formal logic the concepts "contradictory" and "logically false" are used synonymously, and at least nominally there is a clear contrast to Hegel's views. However, does formal logic have a better answer to the question of what logical falsity is than to the question of what contradiction is? In truth, it is not the task of formal logic to discuss such questions, let alone to investigate whether there is something in the objects of contradictory or tautological judgments that "corresponds" to contradiction or tautology.

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Hegel assigns this task to a "science" that makes logic itself its subject mat ter: the "science oflogic."

NOTES

1. Originally published as "Uber Hegels Lehre vom Widerspruch," in Dieter Henrich, ed., Probleme der Hegelschen Logik (Stuttgart: KlettCotta, 1986), pp. 107-28.

2. Hegel's Science of Logie, trans. A. V. Miller (New York: Humanities Press, 1969; cited as "SL"), pp. 431-43. (Miller's translations have been revised without notice.) Wissenschaft der Logik, ed. G. Lasson (Hamburg: Felix Meiner, 1975; cited as WL), vol. II, pp. 48-62.

3. Hegel's Aesthetics, trans. T. M. Knox (Oxford: Clarendon Press, 1975), vol. 2, p. 194: "The moving emotion is the sense of the dialectical contradiction of having sacrificed one's personality and yet of being independent at the same time, this contradiction is ever present in love and ever resolved in it." The expression "dialectical contradiction" apparently be came an established philosophical term only after Lenin. I have not found it in Marx or Engels. In Lenin's work there is perhaps only one reference ("Uber das Verhaltnis der Arbeiterpartei zur Religion" [1909] in Werke [Berlin: Dietz, 19621, vol. 15, p. 409), to which Prof. E. Albricht (Greifswald) kindly drew my attention.

4. I attempt to provide such a commentary in my book Der Begriff des Widerspruchs. Eine Studie zur Dialektik Kants und Hegels (K6nigstein/fs.: Hain, 1981).

5. Cf. Kant's discussion of the socalled Antinomies in the Critique of Pure Reason. The Antinomies are pairs of judgments that have a contradictory form, although their contents are either contrary or subcontrary.

6. See esp. "The Highest Principle of all Analytic Judgments" in Kant's Critique of Pure Reason (AI5053/BI89-93). Cf. M. Wolff, "Der Begriff des Widerspruchs in der 'Kritik der reinen Vernunft,'" in B. Tuschling, ed., Probleme der Kritik der reinen Vernunft, (Berlin: de Gruyter, 1984), pp. 178-226.

7. W. V. O. Quine, "Two Dogmas of Empiricism," in From a Logical Point of View (New York: Harper, 1953), p. 20.

8. Cf. the introductory remarks in Der Begriff des Widerspruchs (op. cit.). Diderot's physi. ological works are of particular interest in this regard.

9. This fact is pointedly expressed in an early formulation by Hegel: "To recognize the principle of contradiction as a formality, thus means to cognize its falsity at the same time." "Relationship of Skepticism to Philosophy," trans. H. S. Harris, in G. di Giovanni and H. S. Harris, eds., Between Kant and Hegel, (Albany: SUNY Press, 1985), p. 325.

10. Hegel's theory of movement and his associated assessment of Zeno's paradoxes have been largely misunderstood heretofore. This is true especially of the critique of Hegel in spired by Cantor's set theory, going back to B. Russell. For details see Der Begriff des Widerspruchs (op. cit.), chap. 1.

11. Cf. A. Trendelenburg, Die logische Frage in Hegels System. Zwei Streitschriften (Leipzig: Brockhaus, 1843), p. 15; E. V. Hartmann, Ober die dialektische Methode (Berlin: Duncker, 1868), p. 72 ff.; Chr. Sigwart, Logik (Tubingen: Mohr, 1873), vol. I, p. 128; G. Patzig, "Hegels Dialektik und Lukasiewiczs dreiwertige Logik," in Das Vergangene und die Geschichte (G6ttingen: Vandenhoeck & Ruprecht, 1973), pp. 443-60; idem, "Widerspruch," in H.

Hegel's Doctrine of Contradiction 21

Krings, et aI., eds., Handbuch philosophischer Grundbegriffe (Munchen: Koesel, 1974), vol. 6, p. 1679f.; W. Stegmuller, HauPtsriirnungen der Gegenwartsphilosophie (Munchen: Kr6ner, 1975), vol. 2, p. 142.

12. A more detailed exposition of this doctrine, not systematized by Kant himself but developed in various parts of his main works, is found in part 1 of Oer Begriff des Widerspruchs (op. cit.).

13. The concept of a logic of reflection is explained further below with some specific cases. For detailed discussion, see Oer Begriff des Widerspruchs (op. cit.). Here the following remark must suffice: traditional logic of reflection concerns logical relations between concepts; more exactly, between intensions of concepts. According to Hegel, who transformed the logic of reflection into an "objective logic," one should rather say that it concerns logical relations between determinations.

14. Hegel presented this law in his second remark to the section on contradiction and in 119 of his Encyclopedia in the following, illunderstood, symbolic form: "to A belongs either +A or - A." "+A" and" - A" are variables for names of predicates, which by themselves are taken neither as contrary nor as contradictory. Only in the context of a complete judg ment do these predicative components constitute an opposition, as must be presupposed by the contradictory relation of two predications. The unsigned "A" does not denote anything whatsoever; instead it denotes the substrate of logical reflection. The common letter "A" in the predicate name and in the subject term expresses the material, logically reflective depen. dence of the meaning of the subject term on the opposition relation of predications. For clarification see below.

15. "Attempt to Introduce the Concept of Negative Magnitudes into Philosophy," trans. D. Walford and R. Meerbote, in l. Kant, Theoretical Philosophy: 1755-1770 (Cambridge: Cambridge University Press, 1992), pp. 203-41.

16. Dieter Henrich, "Hegels Logik der Reflexion. Neue Fassung," in Oie Wissenschaft der Logik und die Logik der Reflexion; HegelStudien Beiheft 18 (Bonn: Bouvier, 1978), p. 261.

17. F. Cajori, Arithmetic, Algebra, Zahlentheorie, in Moritz Cantor, ed., Vorlesungen uber Geschichte der Mathematik (l st ed., 1908, repr. New York-Stuttgart: Johnson, 1965), vol. 4, pp.79-89.

18. A. G. Kastner, Anfangsgrunde der Arithrnet ik (G6ttingen: Vandenhoeck & Ruprecht, 1758). 19. See the appendix to the third part of the Transcendental Analytic: "The Amphiboly of

Concepts of Reflection" (A260-92/B316-49). 20. G. S. Klugel, "Entgegengesetzte Gr613en," in Mathernatisches Wiirterbuch, 1 st div., 2nd

pt. (Leipzig: Schwickert, 1805). 21. Cf. E. Landau, Grundrege/ der Analysis (1930; repr. Darmstadt: Wissenschaftliche

Buchgesellschaft, 1970), pp. 76, 85. On H. GraBmann's innovation, see his "Snicke aus dem Lehrbuche der Arithmetik" (Berlin: Enslin, 1861), 4, p. 54: "Explanation," in: Gesarnrnelte rnathernatische und physikalische Werke (Leipzig: Teubner, 1904), vol. 2.1, p. 314.

22. Hegel uses the unsigned "a" in place of the modern symbol" I a I." " I a I" originated as the abbreviation of ",fa'."

23. J. W. von Goethe, Goethe's Theory of Colours (London: Murray, 1840; repro Cllnbridge, MA: MIT Press, 1970).

24. One should compare this somewhat bold, if not apparently mystical, remark with Wittgenstein's Tractatus, 4.461: "Propositions show what they say: tautologies and contra dictions show that they say nothing." In 4.4611 the parallel becomes still clearer when

22 The Owl of Minerva 31:1 (Fall 1999)

Wittgenstein (if only obliquely) relates the concept of contradiction to the concept of zero. Hegel develops a more detailed theory of this relation in the section on "Contradiction" in the Science of Logic. However, the opposition between Wittgenstein and Hegel should not be overlooked. See Tractatus, 4.462: "Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit all possible situations, and the latter none." Ludwig Wittgenstein, Tractatus Logico.Philosophicus, trans. D. F. Pears and B. F. McGuinness (London: Routledge & Kegan Paul, 1961).