Willem van der Kuijlen. Infinite judgment in Kant’s «Critique of pure reason»


Willem van der Kuijlen

1. Introduction

In the table of logical functions of understanding Kant introduced «infinite judgments» in addition to the common logical distinction between affirmative and negative judgments[1]. Correspondingly, Kant included «limi­tation» in the table of categories in addition to the categories of «reality» and «negation» (KrV, B106). Kant’s example of infinite judgment was the sentence, «The soul is non-mortal». Kant’s division of judgments may come as a surprise to those familiar with ordinary logical distinctions, especially because, in general logic, infinite judgment is classed with affirmations (KrV, B96f.). In view of transcendental logic, therefore, Kant took care to distinguish it from both negative and affirmative judgments. It is not negative because it is a logical affirmation, since the copula is not negated. Unlike ordinary affirmative judgments, on the other hand, it contains a negative predicate. Hence, infinite judgment is a logical affirmation by using merely a negative predicate in order to say something about our total knowledge (KrV, B97) and the content of knowledge «überhaupt»[2].

Notwithstanding this explanation, Kant’s introduction of infinite judgment and the category of limitation has met with neglect and misunderstanding, if not downright rejection, ever since. Schopenhauer, as always, was clear and quite amusing in his disapproval. He called infinite judgment «einen spitzfindig erdachten Lückenbüßer, was nicht ein Mal einer Auseinandersetzung bedarf, ein blindes Fenster, wie er zu Gunsten seiner symmetrischen Architektonik deren viele angebracht hat.»[3]. According to Hegel it was not a judgment at all[4]. Peirce also maintained that Kant added infinite judgments «because it rounded out his triad of categories of quality».[5] As to the category of limitation, De Vleeschauwer added a philological reason: «Elle constitue donc une retouche de la derničre heure.»[6]. To Kemp Smith, Kant’s distinctions reflected «a very artificial and somewhat arbitrary manner» in preparation «for the ‘discovery’ of the category of limitation»[7].

Instead of blaming Kant for unclarity or architectonical preoccupations to account for misunderstandings on our own part, Kant’s distinction itself should be taken more seriously. The following sections will examine Kant’s notion of infinite judgment, its meaning and function in view of the evidence found throughout KrV instead of particular passages only. The methodological starting point of this analysis is the assumption that Kant must have had good reasons to introduce infinite judgment; otherwise he would not have mentioned it at all and he certainly would not have added his warning observation in KrV, B97f..

Kant’s insistance on this form of judgment is not as far-fetched as his commentators would have it. In ordinary language it is not uncommon to use negative predicates and even double negations in a meaningful way. Especially cases of double negation may help us to understand the meaning and function of infinite judgment. Generally, double negation amounts to an affirmation of what is negated: duplex negatio affirmat. This may apply in the case of classical bivalent logic, which is the logic most philosophers tend to subscribe to, but if it is considered as a principle of logic (sc. the law of double negation[8]) it is not appropriate to serve as a logical means to analyse occurrences of double negation in ordinary or non-formal language. For example, the use of figures of speech such as understatement and litotes[9], which may contain double negations, cannot be adequately understood if they are supposed to represent affirmations. In order to account properly for occurrences of double negation, we must assume that there are good reasons for using the long-winded expression instead of a shorter one[10]. These reasons may be of a pragmatic, rhetoric, or political nature, but in each case we will have to take the context into consideration.

From the following example it will be clear that double negation does not amount to a simple affirmation. Sentence 1 is a statement in response to official claims that there was no point in arresting Bouterse in Brazil, because subsequent extradition to The Netherlands would be impossible[11].

1 «The arrest and extradition of Desi Bouterse by the Brazilian government is complicated, but not impossible.»

«Not impossible» in this case does not mean the same as «possible», at least that is not the exact meaning of the statement. The statement merely points to the fact that the opposite claim about the impossibility is not valid. This does not entail a true claim about the possibility of arrest and extradition.

For the sake of introduction it is assumed here that this example should be understood in terms of infinite judgment. Infinite judgment, then, seems to be a highly common means to express specific meanings in specific contexts for specific reasons[12]. As we shall see, the topics related to this example (esp. double negation) will recur when infinite judgment in the Kantian sense is discussed.

In the next section we will have a closer look at infinite judgment in relation to transcendental knowledge. In section 3 we will assess historical and systematic objections raised against infinite judgment. Implications resulting from this evaluation are presented in section 4 in relation to Kant’s conception of transcendental logic. Sections 5 and 6 will determine the meaning and function of infinite judgment in the transcendental dialectic and doctrine of methods of KrV.

2. Infinite judgment and transcendental knowledge

In his observation regarding infinite judgment (KrV, B97f.) Kant clearly distinguished it from affirmative and negative judgment. His example was «The soul is non-mortal.», in contrast with the negative judgment «The soul is not mortal.». We may rephrase the latter example as «It is not the case that the soul is mortal.» to show that negative judgment is the negation of an affirmative one. Although Kant did not give a corresponding example of affirmative judgment, it obviously would have been «The soul is mortal»[13]. These three examples, the three types of judgment and their formalisations, are represented as follows:

2    The soul is mortal        – affirmative judgment  –    S is p

3    The soul is not mortal  – negative judgment      –    ¬(S is p)

4    The soul is non-mortal – infinite judgment        –    S is ¬p

Kant did not mention an example of affirmative judgment because, in this case, it would have been false: the soul does not belong to the set of mortal things. This implies an important aspect of Kant’s view of judgment. By «judgment» he meant «true judgment». This is confirmed by Kant’s conception of the function of negative judgment. In KrV, B97 he stated that a negative judgment regarding the soul would have warded off error. Negative judgments prevent us from making mistakes (cf. KrV, B737). True infinite judgments, however, serve another purpose.

Kant also provided an explicit reason to introduce infinite judgment: only in transcendental logic is infinite judgment a separate member in the division of judgments. In general logic, however, it is rightly classed with affirmative judgments. Apparently, it is necessary to distinguish infinite judgment because it serves a particular function in transcendental logic. In order to understand this function we have to get a clear picture of how transcendental logic should be distinguished from general logic. The remainder of this section is devoted to the main aspects of this distinction.

General logic abstracts from all content of the predicate. It abstracts from all content of knowledge and deals with nothing but the mere form of thought (KrV, B78)[14]. Transcendental logic, however, also considers what may be the worth or content of a judgment (KrV, B97). Here, «content» should be taken in a very specific way, i.e. in the sense of «Gegenstände a priori» (KrV, B81) or «Erkenntnis a priori» (KrV, B80). Thus, in transcendental logic, content is limited to pure knowledge a priori (KrV, B171). The content matter of transcendental logic does not consist in objects (e.g. bodies) or properties (e.g. mortality) in the ordinary sense, but in knowledge. Transcendental logic expresses knowledge about knowledge.

If we take a closer look at Kant’s explanation of the term «transcenden­tal» this conception of logic is easier to understand. In his introduction to KrV Kant stated:

«Ich nenne alle Erkenntnis transzendental, die sich nicht so wohl mit Gegenständen, sondern mit unserer Erkenntnisart von Gegenständen, so fern diese a priori möglich sein soll, überhaupt beschäftigt» (KrV, B25).

Transcendental logic is a specific kind of transcendental knowledge. The latter concerns the a priori possibility of the mode of knowledge of objects, rather than one of the objects themselves. More specifically, this is stressed right at the beginning of «Die transzendentale Logik»:

«nämlich: daß nicht eine jede Erkenntnis a priori, sondern nur die, dadurch wir erkennen, daß und wie gewisse Vorstellungen (Anschauungen oder Begriffe) lediglich a priori angewandt werden, oder möglich sein, transzendental (…) heißen müsse. (…) Der Unterschied des Transzendentalen und Empirischen gehört also nur zur Kritik der Erkenntnisse, und betrifft nicht die Beziehung derselben auf ihren Gegenstand» (KrV, B80f.).

According to this passage transcendental knowledge belongs to the project of the critique of knowledge, i.e. the critique of pure reason as it is presented in KrV. From this perspective we could distinguish between various levels of knowledge. The most general level is that of critique. Transcendental knowledge, as distinguished from empirical knowledge, is a more specific kind of knowledge. It is not so much concerned with a priori knowledge tout court, but only with a priori knowledge about the a priori possibility or application of certain representations (intuitions and concepts), i.e. the mode of knowledge (as specified in the transcendental aesthetic and the transcendental logic respectively). The third level contains knowledge about what is ordinarily regarded as the content, or subject matter of statements. This knowledge is expressed in statements like «Bodies are heavy». The content referred to in this type of statement is possible experience, which would be yet another level, i.e. the «lowest» content level.

For our present purposes we have identified four levels: 1) objects and properties within the field of possible experience; 2) judgments about 1; 3) transcendental knowledge about the a priori conditions of 2 (i.e. all that Kant said about space and time, the categories, schematism and the principles); and 4) critique as presented in KrV. As Kant stated, and as has been pointed out above, transcendental logic considers what may be the worth or content of a logical affirmation in infinite judgment, and what is thereby achieved in the way of addition to our total knowledge or in respect of the content of knowledge in general («überhaupt»)[15]. Thus, infinite judgment as part of transcendental logic is not a logical distinction for the affirmation or negation of anything at levels 1 and 2. Rather, it is a means to express something about knowledge überhaupt, i.e. about a higher level of knowledge. Seen against this background, infinite judgment is probably the only form of judgment which is specifically fit to put critical knowledge into words. Infinite judgment is indispensable for the purpose and means of Kant’s critique. In the following sections this claim will be substantiated.

3. Against infinite judgment

In his study of Kant’s infinite judgment Albert Menne[16] echoed Schopenhauer’s reproach: Kant introduced infinite judgment to save the symmetry of the table of categories (Menne, l.c., 151f., 159). Menne denied the independent status of infinite judgment for historical and systematic reasons. Historically speaking, there is no justification for using the notion of infinite judgment, since, as Menne suggests, the use of the term «infinite» is based on an incorrect translation of «indefinite judgment» (enuntiatio infinitus), which is a judgment containing a negative or infinite term with an infinite extension. The misunderstanding of Aristotelian «indefinite judgment» probably lead Kant to disregard this kind of judgment as the fourth, quantitative form of judgment and to introduce a missing, third, qualitative form of judgment, thereby completing the trichotomy of the table.

However, within the historical context, Menne acknowledged that some logicians prior to Kant also spoke of «infinite judgment», for example, Scharffius, Micraelius[17] and Jungius (l.c., 155-157), although the use of infinite judgment was not wide-spread and limited only to approaches that were not Aristotelian or scholastic. According to Menne, Kant used an unusual type of judgment in a very specific way as he attached literal meaning to «infinite», although he was ignorant of its proper historical and philosophical background[18]. Besides, Menne also maintained that there were no systematic reasons to use it in the same literal way as Kant did.

Menne (l.c., 159-162) denied Kant’s literal reading of infinite judgment systematic relevance, especially in so far as it is distinguished from negative judgment, because there is no difference between a negative and infinite judgment, if the relations between subject and predicate are represented according to set theory. In agreement with this view negative judgment 3 and infinite judgment 4 are set theoretically identical, if negation in either judgment is understood in terms of complements: «not being mortal» and «non-mortal», both formalised as «(mortal’)», refer to the complements of the set of mortal things. The formalisation expressing this identity would run: {x | soul (x)} ? {x | mortal’ (x)} (Menne gives diagrams). In Menne’s view, therefore, there is no difference between negation of the copula and predicate negation. To support this, he also called upon the generally accepted laws of traditional logic, particularly Apuleius’ laws of equipollence, which state that an affirmative judgment is equipollent to the corresponding negative judgment if the predicate is denied as well[19]. Menne concluded that both kinds of negation should be equivalent in order to yield an affirmative result. Applied to 2 and 3 above this may be expressed as:

5    (S is p)  ¬(S is ¬p)

Menne’s argument is circular (claiming the equivalence and proving it by referring to a defining, common logical principle) and appears to be based on the principle of double negation: two negations (of a single term or within a single statement) equal the affirmation (of that term or statement):

6    A   ¬ ¬A     principle of double negation

Menne’s unconvincing, or at least unsatisfactory rejection of infinite judgment is based on this principle and on the principle of excluded middle, as is clear from his conception of «non-p» in terms of the complement of «p». Thus he maintains that if the introduction of infinite judgment were to make any sense, it would have been necessary for Kant to show that these formal principles do not hold in the case of transcendental logic. According to Menne (l.c., 160) Kant failed to supply such arguments and only offered problematic speculation. I believe that these arguments, apart from the basic distinction between transcendental and general logic (see section 2), can be found quite explicitly in other parts of KrV. A more detailed discussion of infinite judgment in the next section will deal with the principles of double negation, excluded middle and, in addition, the closely related principle of contradiction.

4. Infinite judgment in transcendental logic

An explanation of infinite judgment should be based on an explanation of the applicability of the principles involved, as it does not make sense to introduce infinite judgment if these principles apply without any restrictions. Again, the starting point is Kant’s conception of «transcendental». In his view, not all types of a priori knowledge are transcendental. Analytic judgments are a priori judgments, but they are not transcendental. Because analytic judgments only express the fact that the predicate is contained in the concept of the subject, which is exacly what makes them analytic (KrV, B10), it is not necessary to rely on experience in order to be able to make them (KrV, B11f.).

Reference to experience is what distinguishes analytic from synthetic judgments. Synthetic judgments express a relation between subject and predicate which cannot be derived from the relation between the concepts involved. Synthetic judgments a posteriori can only be derived from experience, e.g. «This body is heavy» (KrV, B12). However, in the case of synthetic judgments a priori (e.g. «Everything which happens has its cause.», KrV, B13) things grow more complicated. Experience can no longer serve to account for the synthetic character of these judgments, since they are a priori. Thus the key problem of the critique of pure reason can be defined as the search for an answer to the question «How are a priori synthetic judgments possible?» (KrV, B19, 73). The answer in terms of the a priori possibility and applicability of concepts and intuitions is what Kant called «transcendental knowledge».

Now we should determine in what transcendental way the principle of contradiction[20] applies to analytic and synthetic knowledge. In KrV, B190 Kant defined the principle of contradiction as the highest principle of analytic knowledge:

7    «Keinem Dinge kommt ein Prädikat zu, welches ihm widerspricht»

In a sense, the principle of contradiction applies to analytic as well as synthetic judgments:

«ein synthetischer Satz kann allerdings nach dem Satz des Widerspruchs eingesehen werden, aber nur so, daß ein anderer synthetischer Satz vorausgesetzt wird, aus dem er gefolgert werden kann, niemals aber an sich selbst» (KrV, B14).

Thus, the principle can be applied to a synthetic judgment, but only in so far as yet another synthetic judgment is presupposed from which it follows. To the latter it cannot be applied (unless in so far as… etc., which would initiate an infinite regress). Again, this limited application of the principle of contradiction indicates a need to account for the possibility of a priori synthetic judgments.

In addition to the principle of contradiction, Kant also defined the highest principle for all synthetic judgments:

8    «ein jeder Gegenstand steht unter den nothwendigen Bedingungen der synthetischen Einheit des Mannigfaltigen der Anschauung in einer möglichen Erfahrung.» (KrV, B197).

This principle contains the very short answer to the question about the possibility of a priori synthetic judgments. It is also a specimen of transcendental knowledge and it should be dealt with in transcendental logic, as Kant emphasised (KrV, B193). The principle of contradiction, on the other hand, «gehört aber… bloß in die Logik» (KrV, B190), thus stressing once more the disregard of content in this type of logic.

In the synthetic a priori, however, content is certainly taken into account, which is indicated in 8 by the occurrence of «Gegenstand» (object). The only objects given are objects of experience, in which case they should be taken as real objects. The principle of the synthetic a priori as stated in 8 specifies that the content of knowledge corresponds to an objective reality if the objects can be given in possible experience, i.e. if they comply with the synthetic requirements of sensibility (space and time), understanding (categories) and unity of apperception. Conformity to these formal, synthetic conditions of experience constitutes the objective reality or validity of knowledge. Synthetic principle 8 concerns content in so far as it specifies the conditions of what is to count as objectively real.

Kant’s reference to «objective reality» (KrV, B194-196) in relation to the principle represents a major indication of how he conceived of the function of the three types of judgment in transcendental logic. In transcendental logic, affirmative judgments are true judgments, representing transcendental knowledge about what is to be considered as objectively real knowledge. This also makes it clear that an affirmative judgment is directly related to the category of «reality»: it affirms at a transcendental level what is valid and real at object level.

If this analysis is basically right, we may draw some tentative conclusions in respect to negative and infinite judgments in transcendental logic. Presumably, at a transcendental level negative judgment determines what should not be counted as a valid or objectively real claim of knowledge. In addition, infinite judgment is an affirmation, which is neither valid, nor invalid. Infinite judgment is non-valid. In the next section these conclusions will be further substantiated. Once again, we will be confronted with the limited applicability of the laws of contradiction and double negation as far as infinite judgment is concerned.

5. Infinite judgment, antinomy and “omnimoda determination”

If the proper function of negative judgment is to ward off error, its position in transcendental logic is implied in our analysis of affirmative judgment in the preceding section. At the level of the Transcendental Dialectic negative judgment is a true judgment about a false affirmative judgment regarding objects which cannot be given in possible experience (the soul, the world, God). This makes the Transcendental Dialectic the negative counterpart of the Transcendental Analytic. In this sense Kant referred several times to the important negative function of critique (KrV, Bxxivf., 25, 740, 823). Infinite judgment, the main focus of attention here, is also relevant in Transcendental Dialectic.

The clearest example of the presence of infinite judgment in the Transcendental Dialectic is Kant’s solution of the antinomy, especially the first antinomy about the (in)finity of the world (KrV, B454-461). Reason gets entangled in the antinomy as soon as it tries to extend knowledge (cosmological ideas) beyond the limits of possible experience. The result is a deadlock; opposing rules seem to apply in the case of transcendent, cosmological knowledge, although they cannot both apply, because that would contradict their rule-like character. Furthermore, there does not seem to be a way out of this situation: antinomy is a fundamental conflict of reason. Kant’s construction of the antinomy in transcendental logic shows the insufficiency of the logical principles involved. We will reconstruct Kant’s analysis of the antinomy taking into account Ishikawa’s study on infinite judgment and antinomy[21].

As in any other case of antinomy, the first antinomy consists of a thesis, supported by an argument to prove it, and an antithesis claiming the opposite of the thesis and also supported by proof. The thesis is, «The world is finite», whereas the antithesis runs, «The world is infinite»[22]. In support of the thesis, Kant constructed the following proof structure. Assume the opposite of the thesis (sc. the infinity of the world) to be true. According to Kant, this assumption would force us to accept contradictions which follow from it. Hence, we must reject this opposite claim because it is false, thus ipso facto accepting the thesis. Elsewhere (KrV, B817-822) this proof structure was called apagogical or indirect: justifying a claim by refutation of the opposite. Kant adopted the same strategy in case of the antithesis.The result, however, is a conflict of reason to which there is no apparent solution (for if there were, it would not be an antinomy).

The antinomy arises due to strict application of the logical principles mentioned above. The false statement about the infinity of the world entails a true statement about the finity (and vice versa in the case of the antithesis), given the principle of double negation (¬¬finity   finity) and the principle of excluded middle (finity    infinity). The result is a contradiction (antinomy, conflict of reason), which poses a serious problem because thesis and antithesis cannot both be true, given the principle of contradiction (¬(finity   infinity)). In Kant’s solution to the latter problem, infinite judgment plays a decisive role.

In the critical solution to the antinomy (section 7 of the antinomy chapter) Kant initially described the first antinomy as an analytical opposition. To solve it, however, this antinomy should be described in terms of a dialectical opposition, which is different from a contradictory opposition (KrV, B532). To do so, Kant employed predicate negation (sc. «nichtunendlich»), which is characteristic of infinite judgment, instead of a negative copula (sc. «non est infinitus») (KrV, B531f.). Then, the antinomy was rephrased as follows: «die Welt ist entweder unendlich, oder endlich (nichtunendlich)» (KrV, B532)[23]. Thus, double predicate negation may also occur in an infinite judgment.

In what way does infinite judgment help to solve the antinomy? Contrary to the antithesis in an analytical opposition, infinite judgment in a dialectical opposition attaches a determination to the world, regarded as a thing actually existing in itself. If the world does not exist as a thing in itself, both thesis and antithesis are false since there is no such thing that is either finite or infinite. Indeed, we cannot regard the world as something existing in itself; it exists only in the empirical regress of the series of appearances (according to the results of the transcendental analytic) (KrV, B532f.). Thus, the solution consists of the introduction of infinite judgment, which enables us to «unmask» the antinomy as an analytical opposition based on the assumption of the world existing in itself.

Although Kant only presents the solution in the case of the first antinomy, it also holds for the other three: «Was hier von der ersten kosmologischen Idee… gesagt worden, gilt auch von allen übrigen.» (KrV, B533; cf. Ishikawa, o.c., 93). As to the mathematical antinomy, thesis and antithesis are both wrong; in the dynamical antinomy, they may both be true (KrV, B556, 559f., 590).

According to Ishikawa there is yet another instance of infinite judgment present in Kant’s solution. Kant’s formulation of the solution has the form of infinite judgment (Ishikawa, o.c., p. 87, 96, 117f.). At this point it is clear that infinite judgment is proper to the perspective of transcendental knowledge, because the solution (or rather: «decision», «Entscheidung», KrV, B525) is part of the critical, transcendental level of knowledge.

In addition to Menne’s historical account there is more to be said about infinite judgment if we look at the beginning of the third chapter of the Transcendental Dialectic. In KrV, B600f., Kant stated that every thing, as regards its possibility, is subject to the ontological principle of complete determination (omnimoda determinatio), «nach welchem ihm von allen möglichen Prädikaten der Dinge, so fern sie mit ihren Gegenteilen verglichen werden, eines zukommen muß.» (cf. also KrV, B596). This principle concerns the content, rather than the mere logical form of knowledge. This feature distinguishes it from the logical determinability of concepts, which is in fact the combination of the principle of contradiction and excluded middle (KrV, B600n.). As we have pointed out above, this distinction is exactly what makes the difference between infinite judgment, on the one hand, and affirmative and negative judgment, on the other.

There is a close link between infinite judgment and the omnimoda determinatio. This principle represents the (historical) origin of infinite judgement (Ishikawa, o.c., 57-69). It also confirms systematic points that were neglected by Menne, which make Kant’s use of infinite judgment meaningful. The main point of agreement between infinite judgment and the principle is that they relate to a content, i.e. a thing. Every thing that exists is completely determined (KrV, B601). In accordance with this proposition specific pairs of contradictory predicates are not only compared with one another logically, but the thing itself is transcendentally compared with the sum total of all possible predicates (ib.). In its turn, the sum total itself is also completely determined as the concept of a thing called the ideal of pure reason (KrV, B601f.). This ideal represents the transcendental content, the transcendental substrate, the matter, the whole of reality or the omnitudo realitatis of the possibility and complete determination of all things.

The complete determination of a thing rests on the limitation of this omnitudo; it is completely determined in so far as the omnitudo is limited to a certain extent or, as Kant puts it «die durchgängige Bestimmung eines jeden Dinges beruht auf der Einschränkung dieses All der Realität» (KrV, B605). It is exactly at this point that infinite judgment is a meaningful means to express a partial step in the process of complete determination. The negation contained in the judgment does not affect the copula nor does it express the non-existence of a thing, since the existence is presupposed as soon as we take up the determination of a thing. Kant referred to this «presupposition» by calling it «transzendentale Bejahung» in contradistinction to a «transzendentale Verneinung», which means «das Nichtsein an sich selbst» (KrV, B602). Infinite judgment refers to a thing while expressing transcendental affirmation, but it also expresses that only one of two contradictory predicates is to be assigned to a thing, thereby limiting the infinite sphere of all possible predicates represented by the omnitudo. In this context it is clear why Kant’s example of the non-mortal soul in KrV, B97f. is not identical to a negative judgment. In «The soul is non-mortal», “the soul” refers to something and this something is determined to the extent that «mortal» is not an appropriate predicate belonging to that thing.

6. Infinite judgment and the polemical employment of reason

Saying something does not entail the existence of what is said. In fact, in the transcendental analytic existing things are limited to whatever can be given in possible experience. In the Transcendental Dialectic, judgments about illusory objects and their presumed existence are unmasked. Still, Kant allowed for some kind of judgment about these objects, although not in theoretical perspective. In order to say something relevant about these «objects» we see them in a practical perspective, as if they were practically real, not objectively real. In this sense infinite judgment plays its major role.

We have to be brief about this «as if» character of objects in a practical perspective. As early as his critical examination of dialectical illusion Kant alluded to the practical use of objects. As to the substantiality and permanence (immortality) of the soul, for example, we cannot claim knowledge because any such claim would transcend the limits of possible experience, «Gleichwohl wird hiedurch für die Befugnis, ja gar die Notwendigkeit, der Annehmung eines künftigen Lebens, nach Grundsätzen des mit dem spekulativen verbundenen praktischen Vernunftgebrauchs, hiebei nicht das mindeste verloren» (KrV, B424)[24]. But although we cannot claim knowledge, we are entitled to postulate a future life. Strictly speaking, this postulate is not legitimate because there is no possible proof to support it; it transcends the possible limits of knowledge. It is not illegitimate, either, for the very same reason. The only way left open is to claim some right to postulate a future life as if the soul were an object, as long as it serves a practical interest. The proper judgment to express this claim is infinite: it is not illegitimate to claim a future life, since this illegitimacy, expressed in a negative judgment, cannot be proven.

Apparently, an interpretation of the critical result in terms of infinite judgment must also allow the occurrence of double negation (e.g. «not illegitimate») in infinite judgment. In the previous section this was already pointed out in the case of Kant’s solution to the antinomy. There are also examples of Kant’s employment of double negation concerning the critical results in view of a practical perspective: the severity of criticism has rendered reason a not unimportant service (KrV, B424); transcendental employment of pure reason prepares the site for building our moral edifices, a task that is not unmeritorious (KrV, B375); we limit the law of the empirical use of understanding, but not to declare the intelligible impossible (KrV, B590) (all italics mine). These are telling examples, but Kant’s systematic employment of infinite judgment and his arguments are found in the Transcendental Doctrine of Methods.

In the chapter on the discipline of pure reason, Kant supplied a negative legislation (discipline) in addition to the positive legislation of the transcendental analytic. In so far as the speculative use of reason is dialectical, transcendental logic is nothing but a discipline (KrV, B824). Hence, a discipline of pure reason supplies a systematic account of the negative function of critique, which was mentioned at the end of section 4 and the beginning of section 5 above: it is «admonitory negative teaching» (KrV, B740).

The section on the discipline of the polemical employment of reason seems to be an exception, since it seems to represent a relapse into a purely polemical situation, i.e. a situation in which opposing claims constitute a conflict of reason. However, conflicts of this kind have already been solved in the transcendental dialectic on the basis of the formal conditions of knowledge. Hence, conflicts and polemics have basically been prevented: «So gibt’s demnach keine eigentliche Polemik im Felde der reinen Vernunft.» (KrV, B784; cf. KrV, B771, 778). Nevertheless, reason may be employed polemically, but only to defend certain propositions against opposite dogmatic denials (KrV, B767). These propositions themselves are also dogmatic, but they are affirmative and they are made in view of the practical interest of reason (KrV, B769f.; cf. KrV, B777). Here we can see the same strategy that Kant employed in his solution to the antinomy. Neither the assertion nor the denial can be demonstrated conclusively. If the assertion is made in respect of the practical interest, however, there is no conclusive proof to support it, but, what is more important, there is also no valid proof of the opposite dogmatic denial. This lack of proof leaves open the very possibility of practical dogmatic assertions and, consequently, these assertions are not illegitimate. In this sense the position of polemical employment of reason is expressed by the double negation of «not illegitmate» in an infinite judgment, which should not be reduced to the simple affirmation of the legitimacy of that claim. An infinite judgment declaring certain claims «not illegitimate» secures the provisional competence or authority in preparation for the practical employment of reason. Its function is exactly to formulate one of Kant’s main objectives: to deny knowledge in order to make room for faith (KrV, Bxxx).

Finally, even this denial of knowledge is expressed in terms of an infinite judgment. In the section on the impossibility of a sceptical satisfaction of reason, following the section on polemical employment, Kant presented his version of the docta ignorantia in the form of critical ignorance expressing the scope and limits of our knowledge. Ignorance in this sense is not a lack of knowledge, but it is affirmative knowledge about what cannot be known. To be critically ignorant one needs fundamental knowledge of what can and what cannot be known. This ignorance delimits the room available for practical assertions and faith.

7. Conclusion

Infinite judgment is not insignificant. In fact, it is the most important kind of judgment in KrV, since it has the particular properties that are necessary to express the results of Kant’s critical project. Apart from the fact that the distinction between affirmative, negative and infinite judgments corresponds to the structure of KrV (viz. Transcendental Analytic, Dialectic and Doctrine of Methods), Kant employed infinite judgment in his solution to antinomy and paralogisms and in the critical limitation of knowledge which prepares for the practical employment of reason. Furthermore, infinite judgment is used in polemical and defensive strategies to ensure realisation of the practical interest of reason. Neglect of the structural and systematic importance of infinite judgment would pose insurmountable problems to any interpretation of Kant’s critical work.


This article was firstly published in collected articles “Logical Kant-Studies-4” (1998):
Willem van der Kuijlen. Infinite judgment in Kant’s «Critique of pure reason»// Proceedings of the International Workshop “Logical Kant-Studies-4”. Kaliningrad, 1998. P. 199 – 217.

[1] KrV, B95. All references to Kritik der reinen Vernunft (KrV) are in accordance with the second edition of 1787 (= KrV, B). Other references to Kant’s works refer to the Akademie-Ausgabe (= AA, containing volume and page numbers). English translations of KrV are taken from the edition by N. Kemp Smith, New York 1965 (1929).

[2] KrV, B98. Here Kemp Smith’s translation of «Erkenntnis überhaupt» is not adequate. He interprets «überhaupt» in the sense of «only», thereby attaching it to «limitative» («beschränkend») (and adding «their»), whereas, in my opinion, «in Ansehung des Inhalts der Erkenntnis überhaupt» is comparable to «in Ansehung des gesammten Erkenntnisses» (KrV, B97).

[3] A. Schopenhauer. «Kritik der Kantischen Philosophie», in: Die Welt als Wille und Vorstellung I (Zürcher Ausgabe bd. II), Zürich 1977, 559f.

[4] G. Hegel. Wissenschaft der Logik II. Die subjektive Logik (1816), in: G. W. F. Hegel, Gesammelte Werke Bd. 12, Hamburg 1981, p. 70. Nevertheless Hegel states: «Ein reelleres Beispiel des unendlichen Urtheils ist die böse Handlung.».

[5] Ch. S. Peirce. Elements of Logic, sec. 376, in: Hartshorne & Weiss (Eds.), Collected Papers of Charles Sanders Pierce, Cambridge MA 1960, Vol. II, 223. Cf. ib. 227-229.

[6] H. J. de Vleeschauwer. La deduction transcendentale dans l’oeuvre de Kant (t. I), Antwerpen/Paris/’s-Gravenhage 1934-1937, 231. Cf. ib. t. II, 55-57.

[7] N. Kemp Smith. A Commentary to Kant’s Critique of Pure Reason. Atlantic Highlands, 1993 (1918). 192.

[8] On (the law of) double negation cf. Lawrence Horn. A Natural History of Negation, Chicago/London, 1989. P. 22, 296-308.

[9] Cf. Ton van der Wouden. «Litotes and downward monotonicity», in: H. Wansing (ed.). Negation. A Notion in Focus. Berlin/New York, 1996. 145-167.

[10] Cf. Horn’s principle of the division of pragmatic labour: The use of a longer, marked expression in lieu of a shorter expression involving less effort on the part of the speaker signals that the speaker was not in a position to employ the simpler version felicitiously» (Horn, o.c., p. 304).

[11] Cf. De Volkskrant (August 26, 1997, p. 7) reporting on attempts made and not made by the Dutch government to prosecute the former Surinam head of state, Bouterse, for drugs-related crimes.

[12] Ishikawa is not right in believing that infinite judgment does not seem to make sense in ordinary language, cf. Fumiyashu Ishikawa. Kants Denken von einem Dritten. Das Gerichtshof-Modell und das unendliche Urteil in der Antonomienlehre. Frankfurt am Main/Bern/New York/Paris, 1990. P. 34, 99, 100.

[13] It is mentioned in R 3063; «anima est mortalis».

[14] Cf. KrV, B77-79, 83f., 171, 390 and Logik, AA IX, 101.

[15] KrV, B97, 98. Cf. note 2 above. Regarding the limitative function of transcendental logic in relation with formal logic, cf. V. Bryushinkin, «The Interaction of Formal and Transcendental Logic», in: H. Robinson (Ed.), Proceedings of the Eighth International Kant Congress Memphis 1995 (Vol. I-2), Milwaukee 1995, 553-566.

[16] Albert Menne. Das unendliche Urteil Kants, in: Philosophia naturalis 19 (1982), 151-162.

[17] Menne (l.c., 156) referred to Micraelius’ definition of infinite affirmation, but elsewhere Micraelius sharply distinguished between infinite judgment proper and indefinite judgment, cf. J. Micraelius. Lexicon Philosophicum. Stetini, 1662 (1653), 618.

[18] Menne completely ignored what was definitely an important historical source of Kant’s use, sc. Meier’s Auszug aus der Vernunftlehre (1752), §294 (in: AA XVI, 635f.), which was used by Kant in his academic courses since 1756. Kant wrote down several reflections about this section, sc. R 3062 (probably dating back to 1752-1756) through 3072. In R 3069 Kant dealt with judicium indefinitum in the context of limitation and the infinite sphere outside the sphere of a concept.

[19] Menne, l.c., 160. Cf. Horn, o.c., 25. Meier (o.c., §294) also regarded predicate and copula negation as equivalent.

[20] Kant dealt with this principle and its relation to the principle of double negation as early as his Nova dilucidatio (AA I, 389/17-32).

[21] Fumiyashu Ishikawa, cf. note 12 above.

[22] Kant’s phrases are different, sc.: «Die Welt hat einen Anfang in der Zeit …», and «Die Welt hat keinen Anfang …, sondern ist …, unendlich.». In his exposition of the proof of the thesis he used phrases such as «unendlich» and «nicht unendlich», which has led me to employ the shorter phrases.

[23] This formulation is the reverse of the original formulation in KrV, B454f.. This does not imply a different view, cf. Ishikawa, o.c., p. 91, 111.

[24] Cf. also KrV, B166n., 421, B431f. In the antinomy chapter the practical view is dealt with in the section on the interest of reason.